Each layer is more explosive in terms of growth, but calling it "exponentially more" is really understating it. The layers look like this:
Layer 1 is exponentiation: e.g. taking x to x^x.
Layer 2 is repeating Layer 1 x times, each time feeding the new number back into the Layer 1 function. So we get about x^x^x^...^x, where there are x exponents in the exponential tower. Note that this is x^(x^(x^(...^(x^x)...))), not (...(((x^x)^x)^x)...); the former is MUCH larger than the latter.
Layer 3 is repeating Layer 2 x times. So we get
x^x^x^...^x, where the number of exponents in the tower is
x^x^x^...^x, where the number of exponents in the tower is
x^x^x^...^x, where the number of exponents in the tower is
...
repeat x times
...
x^x^x^...^x, where the number of exponents in the tower is x.
So we can no longer even simply describe Layer 3 as a tower of exponents, we need to keep applying the procedure to describe the final number.
Layer 4 is Layer 3 repeated x times, so we get
N(0) = the number we get when we stick in x into the Layer 3 procedure described above
N(1) = the number we get when we stick in N(0) into the Layer 3 procedure described above
...
N(x) = the number we get when we stick in N(x-1) into the Layer 3 procedure described above
So we cannot easily describe a Layer 4 number in terms of the Layer 3 function, we need to iterate it x times to describe it.
And the pattern continues for higher and higher layers. The Knuth arrow notation is defined the same way, so Layer X gives you a function with X Knuth arrows, e.g. Layer 5 takes X to X^^^^^X.
That's the reason that we didn't just use regular math operations to define our numbers, our numbers are much to big to be described with normal notation, even exponentiation. So we had to use Knuth arrow notation to get a handle on these numbers.
As for stages... no, stages really are much more powerful than any number of layers, it's not just more efficient. Sure, if we just apply the stage procedure once with, say, 10 Metallurgeons, we're going to get 22 layers. So if we finish there and call it a day, that's a 22-layer deck. But, what if we repeat the stage procedure?
For example, iet's say we start with 10 untapped Metallurgeons, resolve the stage procedure, then cast Battle Cry, and copy it 8 times. Applying the procedure will get us roughly 10^^^^^^^^^^^^^^^^^^^^^^10 Psychic Battles. Let's switch to Conway notation: we use a -> b -> c to represnt a^^^...^^^b with c arrows. So the previous number is 10 -> 10 -> 22. We have that many Psychic Battles, and also about that many tapped Metallurgeons, since we are creating and tapping new Metallurgeons throughout our whole process. So, the first copy of Battle Cry gets us 10 -> 10 -> 22 untapped Metallurgeons, and now we can reapply the procedure to create 10 -> 10 -> (10 -> 10 -> 22) tapped Metallurgeons. Now that is an enormous number - the number itself has so many arrows, that we need to use 22 arrows to describe the number of arrows it has! There is simply no way any layer deck can get that many arrows. Our current record for a layer deck is 417 arrows for 60 card Vintage deck. For a 100 card Commander deck we are in the mid 600s. So we're never going to get to a million layers or a googolplex layers, much less 10 -> 10 -> 22 layers! And that's with just one copy of Battle Cry. The second copy takes us to 10 -> 10 -> (10 -> 10 -> (10 -> 10 -> 22)). Now this is a number so large, that the number of arrows required to describe it is itself a number so large that the number of arrows required to describe it is itself a number so large that it requires 22 arrows to describe it! Then the third copy of Battle Cry takes us to 10 -> 10 -> (10 -> 10 -> (10 -> 10 -> (10 -> 10 -> 22))), and so on. We get an additional nesting of arrows for each copy of Battle Cry. With 63 copies of Battle Cry, we'll beat Graham's number!
And that's just 1 stage + 1 layer. Basically 1 stage + 1 layer with input X gets you the number
X^^^...^^^X, where the number of arrows is
X^^^...^^^X, where the number of arrows is
X^^^...^^^X, where the number of arrows is
...
repeat X times
...
X^^^...^^^X, where the number of arrows is X.
So if we go to 1 stage plus 2 layers, that represents the procedure we get by repeating the above procedure X times, where each time we set X to the new bigger number we obtained in the previous procedure. Then 1 stage + 3 layers is 1 stage + 2 layers repeated X times, etc.
Then we get to 2 stages, which turns X into 1 stage + X layers. Again, that doesn't seem like a big advance, but you realize that it is once you get to 2 stages + 1 layer. 2 stages + 1 layer repeats the 2 stage procedure X times, that is, we repeatedly apply the procedure 1 stage + X layers, each time feeding the new incredibly large number we get into X. So 2 stages + 1 layer will be bigger than any deck we can create with 1 stage + any number of layers. Then we go to 3 stages, 4 stages, etc.
Hmm, I don't know what you mean by y(x^x) and (x^x)^x, it doesn't look right - (x^x)^x is just Layer 1, since Layer 2 is already x^x^x^...^x with x exponents.
I see what you mean by Harness the Storm. Putting a second copy in is no big deal anyway.
I guess you could use Sleight of Mind or one of the variants to change the color that triggers RSK, but that winds up allowing us to get red mana off of any spell, just like Inexorable Tide. I don't think there is any spell that changes mana symbols; There are some cards that allow for color changing, like Farrelite Priest.
Oh right, with Harness the Storm the original instant/sorcery will still be on the stack when the Harness the Storm triggers get resolved. So that requires either a second copy, or countering the original spell. I guess having a second copy will free up more options, since being able to counter spells will limit what you can do with your spell chain; for example, you could add Timetwister to your deck, but being able to counter Timetwister would make it go infinite. But yeah, two other options are countering, and milling the Surreal Memoir.
Heh, goes to show that I'm narrowly thinking about this challenge and not about whether a card is reasonably powered; obviously Electrodominance would be too good the way I was thinking. I'll try to cook up a turn 6 or turn 7 start.
A stage is process that can take X of a resource and turn it into X layers; for a long time, this was just a pipe dream. I actually stumbled upon the first such combo quite by accident. There are basically just two ways to make a stage process that we know of. One is the Bloodbond March / Psychic Battle combo that I found; tstorm then found a variant where Bloodbond March is replaced by Possibility Storm. The other is the complicated combo created by Stakfish in his Graham's number-beating Standard deck. It really is rather intricate, and the details can be found at his website. The Bloodbond March version is simpler, and appears to be more conducive to creating many stages. The post where I first described it is here. I guess I can try to make a more updated description now.
So, like I said, the key parts are Bloodbond March and Psychic Battle. (Psychic Battle can be replaced or augmented by Grip of Chaos, and in some cases by Rings of Brighthearth.) We have X of some resource that we are going to turn into 2X layers; the resource can be either mana of some color, or some type of creature token. (Credit goes to Iijil for discovering how to use tokens as the resource rather than mana) For this example, let's use Metallurgeon tokens, which are what is used as the primary stage in our megastage and gigastage decks.
So we have X hasted Metallurgeon tokens, and at least X white mana. We also have the
cards on the battlefield, and we have managed to create a bunch of token copies of all our stuff, in particular multiple copies of Psychic Battle and Bloodbond March.
So, we first cast our nontoken Metallurgeon from our hand. This triggers all the Bloodbond Marches. They go on top of the Metallurgeon spell though, so put the Cephalid Shrine trigger on top of the Bloodbond March triggers, and use it to counter the Metallurgeon spell, sending it to the graveyard. We now use the topmost Bloodbond March trigger to bring the Metallurgeon to the battlefield. So now we have Metallurgeon on the battlefield, and a bunch of Bloodbond March triggers on the stack.
Next, before we resolve any of the remaining Bloodbond March triggers, we tap a Metallurgeon token, targeting the nontoken Metallurgeon. This activates multiple copies of Psychic Battle, Cowardice, and Horobi, Death's Wail triggers.
Now, when we have both Cowardice and Horobi, Death's Wail in play, any time we target anything both will trigger. Since the triggers come into existence simultaneously, we can order them on the stack however we so choose. The one that we put on the top is the one that will be executed; since the target of the ability becomes a new permanent when it changes zones, all of the other abilities will fizzle. What this means is, whenever we target any creature, we have the option of either bouncing it back to our hand, or destroying it and sending it to the graveyard, our choice. (If we want the original ability to take effect, we need to remove both Cowardice and Horobi, by the way.)
So, we will put a Cowardice trigger on top, and bounce the nontoken Metallurgeon back to our hand. On the stack is a bunch of Bloodmarch triggers on the bottom, and a bunch of Psychic Battle triggers above those (intermixed with a bunch of Cowardice and Horobi triggers that don't matter anymore, so we will ignore them).
Now, we repeat the process: We replay the Metallurgeon and counter it, getting a bunch of Bloodbond March triggers above our previous bunch of Psychic Battle triggers. Then we tap a second Metallurgeon token to bounce the Metallurgeon back to our hand, and create a second bunch of Psychic Battle triggers above the second bunch of Bloodbond March triggers. We continue this process until we run out of tokens to activate.
After all is said and done, the stack looks like this:
(first to resolve)
group of Psychic Battle triggers
group of Bloodbond March triggers
... (repeat X times)
group of Psychic Battle triggers
group of Bloodbond March triggers
(last to resolve)
and we have the nontoken Metallurgeon on the battlefield. So now the stage is set up, and it's time to start resolving it.
The first thing we resolve is the top group of Psychic Battle triggers. For all but one of those triggers, we will use them to bounce other animated artifacts, in order to make more enchantments and other stuff. If we bounce Azorious Keyrune, we can replay it, and it gets Dual Nature triggers. We can spend three mana to cast Allay with buyback, and that allows us to bounce many enchantments thanks to all the Psychic Battles in play. We use one to bounce March of the Machines, and the rest can bounce Copy Enchantment to keep replaying, doubling the number of Dual Natures each time. And with March of the Machines gone, we can tap all of our Keyrunes to get a bunch of mana, which can be used to cast Allay with buyback many times. So we get a couple of layers already, but this doesn't mean too much; it just means our stage is worth 2X+2 layers or so rather than 2X, which is no longer a big deal at this level. Of course, with our last playing of Copy Enchantment we will copy Psychic Battle.
We have saved one Psychic Battle trigger from our Metallurgeon token, and that is to target Metallurgeon, except instead of putting Cowardice on top of the stack and bouncing the Metallurgeon, we will put the Horobi trigger on top of the stack and send it to the graveyard. This triggers our Mimic Vats; we will imprint the Metallurgeon on one, and tap it to create a hasted Metallurgeon token. After than, we can use Mirror of Fate to put it into our library, and use Perpetual Timepiece to put it into our graveyard. But that's as far as we can get with it; it's stuck in our graveyard until we can get to a Bloodbond March trigger. For this reason, each stack of Psychic Battle triggers can only create one token copy of Metallurgeon, because we need to resolve a Bloodbond March trigger to get it back. This is important, since otherwise we could keep increasing our number of tokens and going infinite.
So, next we resolve the topmost Bloodbond March trigger of the top BM group, and bring it back to the battlefield. We can now tap our newly created Metallurgeon token, and we get Psychic Battle triggers, but now we get way more Psychic Battle triggers than before, because we created many more Psychic Battles while resolving the previous group of Psychic Battles. So our stack differs from before in that our top PB group has way more triggers than before, and our top BM group has decreased by one.
We do the same thing again; each time, the top PM group increases enormously, and the top BM group decreases by one. Eventually, the second BM group will run out. At this point, we do _not_ activate our lone Metallurgeon token! That would create another PB group above the second PB group, with no BM group in between. So our number of groups will have decreased by two, which will cause our whole combo to fizzle out faster than it should. Instead, we save our Metallurgeon token, and resolve the topmost PB trigger from the second PB group, to bounce the Metallurgeon. Then we replay the Metallurgeon, and get many more Bloodbond March triggers than we had before (since we remembered to create more Bloodbond Marches just before this step). Then we use the top BM trigger to bring the Metallurgeon back to the battlefield, and now we tap our token to create more PB triggers.
So now our stack is different in that the first PB group and the first BM group are way bigger than before, but the second PB group has decreased by one.
You can see how this continues. We will keep resolving our top PB group and top BM group, and after they are fully resolved we lose a trigger from the second PB group. When that second PB group is fully resolved, we dip into the second BM group to create a much larger second PB group, and so on down the line.
Eventually, we will completely run out of all of our triggers. We will still have hasted Metallurgeons, since as we resolve down the stacks we build our token reserve back up, but now our Metallurgeon is on the battlefield. So we can build our stack back up, but this time the bottom Bloodbond March group is gone, so we have a stack of 2X-1 groups rather than 2X. Then, after the 2X-1 stack is fully resolved, we can build up a 2X-2 group, and so on, until finally the stack is cleared, and all our untapped hasted Metallurgeons are gone - but we have Ackermann many Psychic Battles and Bloodbond Marches!
Just to count - each time we resolve a Psychic Battle trigger from the top group, we take the number of Psychic Battles from X to about 2^^X. We can resolve X of these before having to dip into the second group, so that takes X to 2^^^X for the whole top PB group. This uses up just one BM trigger from the top BM group, so the whole top BM group takes X to 2^^^^X. Then the second PB group takes X to 2^^^^^X, and so on, until the bottom group is at Layer 2X+2.
So that's it, that's how my version of the stage works. Once you have absorbed this, I can try to help you with hyperstages and beyond. Stakfish has a great summary of the state of things in a post from not too long ago.
Oh yes, I assumed all of that! Basically, having the populate ability, and having turned everything into creatures, means that we can pretty much assume everything will be updated, at the precise right time when it will be most beneficial. So we don't have to worry about some permanent not updating. This is very convenient, but on the minus side, it cuts off all creatures from being a limiter further up the chain, just like Mycosinth Lattice removed different colors of mana and artifacts in particular from being limiters further up the chain. So those are things that you might consider changing up if you _really_ want to push things.
Yeah, Stakfish's deck that exceeded Graham's number went off on turn 6. I think in current Standard we should be able to go off on turn 4, even if we go first. Something like the following:
Draw a card, then attack with all four Merfolk, and draw 12 more cards. With 13 cards in our hand, I don't forsee any further problems; we can first play an Electrodominance for 4. We can play three Runaway Steam-Kins, and then three red spells of any kind, to generate 9 mana; this will allow us to play another Electrodominance for X = 7, which is the most we will need. If we need more cards, we can play Garna, the Bloodflame and Response // Resurgence (these can be two of the red spells that we played) and attack with say another 3 Seafloor Oracles (using Quasiduplicates, perhaps) to draw another 18 cards, then tap them to draw one more. So finishing by turn 4 should be no problem, unless we're really hurting for deck space (which is certainly not the case now!).
I'm pretty happy with turn 4; of course our damage is no where near the Graham's number deck, but I doubt we will be able to replicate that. Maybe a few more layers will be good enough.
Edit: Just saw your last couple of edits. So yeah, I got fixated on the life gain chain and forgot that Wydwen, the Biting Gale would trigger the Inexorable Tides. So we have the casting of a spell at Layer 4, then a resolution of an Abuna's Chant (or whatever life-gaining instant you want to use) will be Layer 5 thanks to Boon Reflection. Then the casting of Abuna's Chant gets layers 6-8 and Surreal Memoir gets layers 9-10. Then I guess I forgot about being able to use Rings of Brighthearth on the clue tokens, so that gets layer 11, and then Press for Freedom gets layers 12-15.
I'm confused by your last edit: Why do we want to get Surreal Memoir into our yard, I thought we wanted to draw it with Clue tokens, and how would Quasiduplicate fix that?
Layer 8: Casting from you hand gets many castings from your graveyard thanks to the Harness the Storm combo
As you can see, the Wydwen / Lobber Crew combo does not add any layers here. Since Lobber Crew just deals damage, it doesn't connect to any of the lower layers, so Wydwen / Lobber Crew are part of their own separate chain that don't lengthen the main chain.
Layers 11-12: Casting Press for Answers gets many copies
Layer 13: Casting Press for Answers from your hand gets many castings from your graveyard
So you're up to layer 13! Not bad at all.
And yeah, you definitely want more Inexorable Tides, since they resolve one at a time, whereas Runaway Steam-Kins will be populated all at once. (They get counters one at a time from red spells, but Inexorable Tides is a teeny bit better even then because you get counters on all RSKs for each resolution rather than just one.)
I'm just going to mention one thing before I analyze the deck further. I just realized that Inexorable Tide just causes spells cast to be at Layer 3, not Layer 4. This is because Inexorable Tide puts counters on all Runaway Steam-Kins at the same time. This means they get affected by exactly the same number of Doubling Seasons, so if you have R Runaway Steam-Kins and D Doubling Seasons, Inexorable Tide resolves as a single step that adds R * 2^D counters in total to your RSKs. This is what we call a "multiplicative effect", and, while that sounds like it is good, what it means is that the layers do not stack; having lots of Doubling Seasons with one RSK would count for a layer, and having lots of RSKs with no Doubling Seasons would count for a layer, but having them both is still just a single layer, it just means that the number to the right of the arrows is bigger, but that advantage will disappear as you go up in layers, as I've tried to show in my previous analyses. So Inexorable Tide is at layer 3.
Contrast that with what happens when you cast a red spell; then, every Runaway Steam-Kin will trigger, and rather than all resolve at once, all those triggered abilities go on the stack to be resolved one at a time. So, you resolve one RSK trigger, and get a whole bunch of counters because of your number of Doubling Seasons. You can then start removing the counters to get red mana and populate, increasing the Doubling Seasons by a lot (at the Layer 2 level). Then, when you resolve the next RSK trigger, there will be a whole bunch more counters created because there are so many more Doubling Seasons in play now. So that is exactly what it means for layers to stack.
So in short, it would be better to go back to Painter's Servant, provided that that does not cause an infinite; if it does, then we will need red spells to get to Layer 4.
And, just as I'm typing this, I realize that you can have a whole bunch of Inexorable Tides, and they will all resolve one at a time. So I'm being silly again. Well, I'll leave it because the above explains why Inexorable Tide is still at Layer 4 and not Layer 5, even though we can add counters to all RSKs at once and not just one.
Some interesting ideas! Giving the opponent a Polyraptor is intriguing, although I don't really know what to do with it that would be better than having it on our side. Maybe some fighting?
The problem with cards like Niv-Mizzet, Parrun is that it will mess up our spell chain, since any card can then fetch any other. To avoid this, we would have to prevent cards going back into our library - but most of our chain aside from Emergency Powers gets cards from the library. It would be nice to find some sort of card drawer that was necessarily restricted in some way - perhaps depending on combat, or the opponent's library.
Electrodominance looks very good, we can preserve the other colors besides red.
Edit: Seafloor Oracle looks quite good. Get 3 or 4 of them out and attack, draw at least 9 or 16 cards.
Edit: Pfft, I'm being an idiot. We don't have Vedalken Orrery! So any sorcery that we try to fetch will be fetched exactly once, then it's stuck in our hand. So we need everything but the last spell in the chain to be instant speed.
Edit: We can extend the chain by one again. This time the base spell is Spit Flame, which we can use to damage a Polyraptor, creating all of one token copy. But that's good enough to start with. Spit Flame can be brought back to our hand when a Dragon comes into the battlefield under our control. So, the next spell up will be Yawgmoth's Vile Offering which can bring a random Dragon back onto the battlefield. YVO is a sorcery, but we can give it flash with Raff Capashen, Ship's Mage. We can destroy our Dragon using a spare copy of Spit Flame, to be brought back again with YVO. YVO then leads into Sphinx's Insight, and then Emergency Powers, as before.
Yeah, the Rowan Kenrith emblems were the second layer. When you activate the populate ability, you get a lot of copies via Rowan Kenrith; each one takes X Doubling Seasons to 2^X Doubling Seasons, so after X copies you get roughly 2^^X. You can use the last ability to create more Rowan Kenriths to update the emblem number.
Yeah, pretty much. If you are using life to pay for additional spell casts, then something that gains you life comparable to the amount of DS/RSK will get you to Layer 5.
Edit: So, I've thought about making a Standard deck a little. Like Stakfish said, Thousand-Year Storm seems like a good card to abuse; we want to set up as long an instant/sorcery chain to make it happen. Ideally, we would like to be able to create many token copies of Thousand-Year Storm, but I don't know of any way to make tokens of enchantments in Standard, and there probably isn't a way.
Obviously, we need a way to get started, a way to generate mana for all our spells, and a way to shuffle cards from our graveyard into our library. So this is just a starting analysis.
Polyraptor and Forerunner of the Empire is a well-known combo; if Forerunner is in play, and we play Polyraptor, we will generate 8 Polyraptors without any interference. More generally, if the Forerunner has X toughness, we will generate about c^X Polyraptors, where c is a little less than 2. So buffing up our Forerunner leads to exponentially more Polyraptors.
Aegis of the Heavens can pump the toughness of Forerunner of the Empire, and with four Thousand-Year Storms, we can get many copies. Casting Ajani's Influence can fetch Aegis of the Heavens many times; if the TYS count is at X, each TYS trigger gets us X copies and allows us to cast Aegis of the Heavens X times, updating the count to 2X. So four triggers will take us to 16X. Expansion // Explosion can fetch Ajani's Influence many times, and one TYS trigger will get X copies, taking X to X*16^X. So four triggers will take X to more than 16^16^16^16^X. Finally, each copy of Flood of Recollection or Vivid Revival can fetch Expansion // Explosion many times, with a single TYS trigger taking X to more than 2^^(4X). With 8 spells and 4 TYS's we get 32 triggers in all, so we will finally reach more than 2^^^35 castings of Aegis of the Heavens. We then get an extra exponent when we count the Polyraptors, but of course that is unnoticeable.
This is just a start, but it looks decent so far.
Edit: Just realized that targeted spells don't play nicely with lots of spell copies entering the stack at the same time. So yeah, we want to replace Flood of Recollection and Vivid Revival with nontargeted spells.
Yep, even with Precursor Golem. adding a token is Layer 1, and Spitting Image aided by Precursor Golem gets two layers in addition to that, so it's at Layer 3, which is insignificant compared to Layer 4.
With the Harness the Storm combo, Spitting Image would get to Layer 4. But even that would make basically no difference. Having two chains at level 4 still takes X to roughly X^^^^X, except that with this both the number of Runaway Steam-Kins and the number of Harness the Storms count. But, it's not even worth it to populate both Runaway Steam-Kin and Harness the Storm; it's much better to populate another Doubling Season, then populate one or the other of RSK and HtS, since that gives you exponentially more. So in fact even X is basically the same here.
Basically, it's really inefficient to have multiple chains up to a particular layer, since you basically get the same benefit with just one, so if you create multiples you wind up using more cards and cutting off more avenues for continuation. So yeah, you should pick one of the two Layer 4 chains and just set up that one.
I know that you like the Spitting Image loop, but you can get the same effect with anything that allows you to return a nonland card to your hand, like Foul Familiar. True, Foul Familiar won't get you the copies that Spitting Image gets, but what you get directly from Spitting Image pales in comparison to what you get from activating all the Runaway Steam-Kins anyway.
@J_kibbs: A good start. With Rowan Kenrith we now have:
Layer 1: Creating a token creates exponentially more thanks to Doubling Season
Layer 2: Activating the Populate ability gets X copies with X Rowan Kenriths
Layer 3: Resoliving a Runaway Steam-Kin triggers gets many counters thanks to Doubling Season
Layer 4: Casting a spell triggers many Runaway Steam-Kins
So if we cast Spitting Image 20 times, along with around 40 other spells once we get the full setup going, we will get to about 2^^^^^62. We can do much better if we set up a spell chain where each spell can cause the previous spell to be cast many times. So for example:
Hmm, it looks like Simian Spirit Guide is no longer safe in my two decks. I was wondering for a minute if it was okay with Iname As One, since Iname doesn't bounce it, but just bringing it to the battlefield allows for a loop. Fortunately, if we remove the SSG stage, the Sea Snidd stage can go right back in. However, this does mean that we don't have enough for the opening hand. So we lose a card slot to put in, say, Mana Crypt. On the plus side, with Mana Crypt we can go back to Words of Wisdom rather than Reforge the Soul.
@J_kibbs: Nicely done, if you couple that with a spell copier, you should get two layers per instant/sorcery that you can chain. I used Harness the Storm in at least one earlier deck, but it's hard to make work because we have Cowardice and Horobi, Death's Wail.
Edit: Thinking some more about the Chalice of the Void deck, I have come up with an idea: What about using Allay instead of Psychatog in order to bounce as well as destroy enchantments, then add Bident of Thassa so we have an artifact that we can repeatedly bounce using all our mana. Then we go to Golem Foundry, where we can use all our artifact casts to create Golem artifact tokens, then have Disenchant, copied using Precursor Golem. As far as I know, Disenchant did not go infinite in our decks, it just wasn't good enough to recreate a stage. Does that work? Perhaps the main problem is that it requires an additional four cards, although Precursor Golem is also useful for the ending layers. So I think that would reduce the deck down to six stages, assuming nothing else changes. Still, it's good to have as many decks fixed as possible, since we don't know what might still go wrong.
Layer 1 is exponentiation: e.g. taking x to x^x.
Layer 2 is repeating Layer 1 x times, each time feeding the new number back into the Layer 1 function. So we get about x^x^x^...^x, where there are x exponents in the exponential tower. Note that this is x^(x^(x^(...^(x^x)...))), not (...(((x^x)^x)^x)...); the former is MUCH larger than the latter.
Layer 3 is repeating Layer 2 x times. So we get
x^x^x^...^x, where the number of exponents in the tower is
x^x^x^...^x, where the number of exponents in the tower is
x^x^x^...^x, where the number of exponents in the tower is
...
repeat x times
...
x^x^x^...^x, where the number of exponents in the tower is x.
So we can no longer even simply describe Layer 3 as a tower of exponents, we need to keep applying the procedure to describe the final number.
Layer 4 is Layer 3 repeated x times, so we get
N(0) = the number we get when we stick in x into the Layer 3 procedure described above
N(1) = the number we get when we stick in N(0) into the Layer 3 procedure described above
...
N(x) = the number we get when we stick in N(x-1) into the Layer 3 procedure described above
So we cannot easily describe a Layer 4 number in terms of the Layer 3 function, we need to iterate it x times to describe it.
And the pattern continues for higher and higher layers. The Knuth arrow notation is defined the same way, so Layer X gives you a function with X Knuth arrows, e.g. Layer 5 takes X to X^^^^^X.
That's the reason that we didn't just use regular math operations to define our numbers, our numbers are much to big to be described with normal notation, even exponentiation. So we had to use Knuth arrow notation to get a handle on these numbers.
As for stages... no, stages really are much more powerful than any number of layers, it's not just more efficient. Sure, if we just apply the stage procedure once with, say, 10 Metallurgeons, we're going to get 22 layers. So if we finish there and call it a day, that's a 22-layer deck. But, what if we repeat the stage procedure?
For example, iet's say we start with 10 untapped Metallurgeons, resolve the stage procedure, then cast Battle Cry, and copy it 8 times. Applying the procedure will get us roughly 10^^^^^^^^^^^^^^^^^^^^^^10 Psychic Battles. Let's switch to Conway notation: we use a -> b -> c to represnt a^^^...^^^b with c arrows. So the previous number is 10 -> 10 -> 22. We have that many Psychic Battles, and also about that many tapped Metallurgeons, since we are creating and tapping new Metallurgeons throughout our whole process. So, the first copy of Battle Cry gets us 10 -> 10 -> 22 untapped Metallurgeons, and now we can reapply the procedure to create 10 -> 10 -> (10 -> 10 -> 22) tapped Metallurgeons. Now that is an enormous number - the number itself has so many arrows, that we need to use 22 arrows to describe the number of arrows it has! There is simply no way any layer deck can get that many arrows. Our current record for a layer deck is 417 arrows for 60 card Vintage deck. For a 100 card Commander deck we are in the mid 600s. So we're never going to get to a million layers or a googolplex layers, much less 10 -> 10 -> 22 layers! And that's with just one copy of Battle Cry. The second copy takes us to 10 -> 10 -> (10 -> 10 -> (10 -> 10 -> 22)). Now this is a number so large, that the number of arrows required to describe it is itself a number so large that the number of arrows required to describe it is itself a number so large that it requires 22 arrows to describe it! Then the third copy of Battle Cry takes us to 10 -> 10 -> (10 -> 10 -> (10 -> 10 -> (10 -> 10 -> 22))), and so on. We get an additional nesting of arrows for each copy of Battle Cry. With 63 copies of Battle Cry, we'll beat Graham's number!
And that's just 1 stage + 1 layer. Basically 1 stage + 1 layer with input X gets you the number
X^^^...^^^X, where the number of arrows is
X^^^...^^^X, where the number of arrows is
X^^^...^^^X, where the number of arrows is
...
repeat X times
...
X^^^...^^^X, where the number of arrows is X.
So if we go to 1 stage plus 2 layers, that represents the procedure we get by repeating the above procedure X times, where each time we set X to the new bigger number we obtained in the previous procedure. Then 1 stage + 3 layers is 1 stage + 2 layers repeated X times, etc.
Then we get to 2 stages, which turns X into 1 stage + X layers. Again, that doesn't seem like a big advance, but you realize that it is once you get to 2 stages + 1 layer. 2 stages + 1 layer repeats the 2 stage procedure X times, that is, we repeatedly apply the procedure 1 stage + X layers, each time feeding the new incredibly large number we get into X. So 2 stages + 1 layer will be bigger than any deck we can create with 1 stage + any number of layers. Then we go to 3 stages, 4 stages, etc.
I see what you mean by Harness the Storm. Putting a second copy in is no big deal anyway.
I guess you could use Sleight of Mind or one of the variants to change the color that triggers RSK, but that winds up allowing us to get red mana off of any spell, just like Inexorable Tide. I don't think there is any spell that changes mana symbols; There are some cards that allow for color changing, like Farrelite Priest.
Heh, goes to show that I'm narrowly thinking about this challenge and not about whether a card is reasonably powered; obviously Electrodominance would be too good the way I was thinking. I'll try to cook up a turn 6 or turn 7 start.
A stage is process that can take X of a resource and turn it into X layers; for a long time, this was just a pipe dream. I actually stumbled upon the first such combo quite by accident. There are basically just two ways to make a stage process that we know of. One is the Bloodbond March / Psychic Battle combo that I found; tstorm then found a variant where Bloodbond March is replaced by Possibility Storm. The other is the complicated combo created by Stakfish in his Graham's number-beating Standard deck. It really is rather intricate, and the details can be found at his website. The Bloodbond March version is simpler, and appears to be more conducive to creating many stages. The post where I first described it is here. I guess I can try to make a more updated description now.
So, like I said, the key parts are Bloodbond March and Psychic Battle. (Psychic Battle can be replaced or augmented by Grip of Chaos, and in some cases by Rings of Brighthearth.) We have X of some resource that we are going to turn into 2X layers; the resource can be either mana of some color, or some type of creature token. (Credit goes to Iijil for discovering how to use tokens as the resource rather than mana) For this example, let's use Metallurgeon tokens, which are what is used as the primary stage in our megastage and gigastage decks.
So we have X hasted Metallurgeon tokens, and at least X white mana. We also have the
Opalescence
Psychic Battle
Cowardice
Horobi, Death's Wail
Bloodbond March
Cephalid Shrine
Mimic Vat
Omniscience
Vedalken Orrery
Mirror of Fate
March of the Machines
Perpetual Timepiece
Dual Nature
Copy Enchantment
Allay
Azorius Keyrune
Metallurgeon
cards on the battlefield, and we have managed to create a bunch of token copies of all our stuff, in particular multiple copies of Psychic Battle and Bloodbond March.
So, we first cast our nontoken Metallurgeon from our hand. This triggers all the Bloodbond Marches. They go on top of the Metallurgeon spell though, so put the Cephalid Shrine trigger on top of the Bloodbond March triggers, and use it to counter the Metallurgeon spell, sending it to the graveyard. We now use the topmost Bloodbond March trigger to bring the Metallurgeon to the battlefield. So now we have Metallurgeon on the battlefield, and a bunch of Bloodbond March triggers on the stack.
Next, before we resolve any of the remaining Bloodbond March triggers, we tap a Metallurgeon token, targeting the nontoken Metallurgeon. This activates multiple copies of Psychic Battle, Cowardice, and Horobi, Death's Wail triggers.
Now, when we have both Cowardice and Horobi, Death's Wail in play, any time we target anything both will trigger. Since the triggers come into existence simultaneously, we can order them on the stack however we so choose. The one that we put on the top is the one that will be executed; since the target of the ability becomes a new permanent when it changes zones, all of the other abilities will fizzle. What this means is, whenever we target any creature, we have the option of either bouncing it back to our hand, or destroying it and sending it to the graveyard, our choice. (If we want the original ability to take effect, we need to remove both Cowardice and Horobi, by the way.)
So, we will put a Cowardice trigger on top, and bounce the nontoken Metallurgeon back to our hand. On the stack is a bunch of Bloodmarch triggers on the bottom, and a bunch of Psychic Battle triggers above those (intermixed with a bunch of Cowardice and Horobi triggers that don't matter anymore, so we will ignore them).
Now, we repeat the process: We replay the Metallurgeon and counter it, getting a bunch of Bloodbond March triggers above our previous bunch of Psychic Battle triggers. Then we tap a second Metallurgeon token to bounce the Metallurgeon back to our hand, and create a second bunch of Psychic Battle triggers above the second bunch of Bloodbond March triggers. We continue this process until we run out of tokens to activate.
After all is said and done, the stack looks like this:
(first to resolve)
group of Psychic Battle triggers
group of Bloodbond March triggers
... (repeat X times)
group of Psychic Battle triggers
group of Bloodbond March triggers
(last to resolve)
and we have the nontoken Metallurgeon on the battlefield. So now the stage is set up, and it's time to start resolving it.
The first thing we resolve is the top group of Psychic Battle triggers. For all but one of those triggers, we will use them to bounce other animated artifacts, in order to make more enchantments and other stuff. If we bounce Azorious Keyrune, we can replay it, and it gets Dual Nature triggers. We can spend three mana to cast Allay with buyback, and that allows us to bounce many enchantments thanks to all the Psychic Battles in play. We use one to bounce March of the Machines, and the rest can bounce Copy Enchantment to keep replaying, doubling the number of Dual Natures each time. And with March of the Machines gone, we can tap all of our Keyrunes to get a bunch of mana, which can be used to cast Allay with buyback many times. So we get a couple of layers already, but this doesn't mean too much; it just means our stage is worth 2X+2 layers or so rather than 2X, which is no longer a big deal at this level. Of course, with our last playing of Copy Enchantment we will copy Psychic Battle.
We have saved one Psychic Battle trigger from our Metallurgeon token, and that is to target Metallurgeon, except instead of putting Cowardice on top of the stack and bouncing the Metallurgeon, we will put the Horobi trigger on top of the stack and send it to the graveyard. This triggers our Mimic Vats; we will imprint the Metallurgeon on one, and tap it to create a hasted Metallurgeon token. After than, we can use Mirror of Fate to put it into our library, and use Perpetual Timepiece to put it into our graveyard. But that's as far as we can get with it; it's stuck in our graveyard until we can get to a Bloodbond March trigger. For this reason, each stack of Psychic Battle triggers can only create one token copy of Metallurgeon, because we need to resolve a Bloodbond March trigger to get it back. This is important, since otherwise we could keep increasing our number of tokens and going infinite.
So, next we resolve the topmost Bloodbond March trigger of the top BM group, and bring it back to the battlefield. We can now tap our newly created Metallurgeon token, and we get Psychic Battle triggers, but now we get way more Psychic Battle triggers than before, because we created many more Psychic Battles while resolving the previous group of Psychic Battles. So our stack differs from before in that our top PB group has way more triggers than before, and our top BM group has decreased by one.
We do the same thing again; each time, the top PM group increases enormously, and the top BM group decreases by one. Eventually, the second BM group will run out. At this point, we do _not_ activate our lone Metallurgeon token! That would create another PB group above the second PB group, with no BM group in between. So our number of groups will have decreased by two, which will cause our whole combo to fizzle out faster than it should. Instead, we save our Metallurgeon token, and resolve the topmost PB trigger from the second PB group, to bounce the Metallurgeon. Then we replay the Metallurgeon, and get many more Bloodbond March triggers than we had before (since we remembered to create more Bloodbond Marches just before this step). Then we use the top BM trigger to bring the Metallurgeon back to the battlefield, and now we tap our token to create more PB triggers.
So now our stack is different in that the first PB group and the first BM group are way bigger than before, but the second PB group has decreased by one.
You can see how this continues. We will keep resolving our top PB group and top BM group, and after they are fully resolved we lose a trigger from the second PB group. When that second PB group is fully resolved, we dip into the second BM group to create a much larger second PB group, and so on down the line.
Eventually, we will completely run out of all of our triggers. We will still have hasted Metallurgeons, since as we resolve down the stacks we build our token reserve back up, but now our Metallurgeon is on the battlefield. So we can build our stack back up, but this time the bottom Bloodbond March group is gone, so we have a stack of 2X-1 groups rather than 2X. Then, after the 2X-1 stack is fully resolved, we can build up a 2X-2 group, and so on, until finally the stack is cleared, and all our untapped hasted Metallurgeons are gone - but we have Ackermann many Psychic Battles and Bloodbond Marches!
Just to count - each time we resolve a Psychic Battle trigger from the top group, we take the number of Psychic Battles from X to about 2^^X. We can resolve X of these before having to dip into the second group, so that takes X to 2^^^X for the whole top PB group. This uses up just one BM trigger from the top BM group, so the whole top BM group takes X to 2^^^^X. Then the second PB group takes X to 2^^^^^X, and so on, until the bottom group is at Layer 2X+2.
So that's it, that's how my version of the stage works. Once you have absorbed this, I can try to help you with hyperstages and beyond. Stakfish has a great summary of the state of things in a post from not too long ago.
Yeah, Stakfish's deck that exceeded Graham's number went off on turn 6. I think in current Standard we should be able to go off on turn 4, even if we go first. Something like the following:
Turn 1:
Play a Stomping Ground and pay the 2 life. Tap to cast a Llanowar Elf.
Turn 2:
Play a Rootbound Crag, tap both lands and the elf to play three more Llanowar Elves.
Tunr 3:
Play Electrodominance with X = 4. Play Sphinx's Insight, and draw two more cards. Play Kumena, Tyrant of Orozca, along with two Seafloor Oracles. Tap all three Merfolk to draw another Seafloor Oracle and play it.
Turn 4:
Draw a card, then attack with all four Merfolk, and draw 12 more cards. With 13 cards in our hand, I don't forsee any further problems; we can first play an Electrodominance for 4. We can play three Runaway Steam-Kins, and then three red spells of any kind, to generate 9 mana; this will allow us to play another Electrodominance for X = 7, which is the most we will need. If we need more cards, we can play Garna, the Bloodflame and Response // Resurgence (these can be two of the red spells that we played) and attack with say another 3 Seafloor Oracles (using Quasiduplicates, perhaps) to draw another 18 cards, then tap them to draw one more. So finishing by turn 4 should be no problem, unless we're really hurting for deck space (which is certainly not the case now!).
I'm pretty happy with turn 4; of course our damage is no where near the Graham's number deck, but I doubt we will be able to replicate that. Maybe a few more layers will be good enough.
Edit: Just saw your last couple of edits. So yeah, I got fixated on the life gain chain and forgot that Wydwen, the Biting Gale would trigger the Inexorable Tides. So we have the casting of a spell at Layer 4, then a resolution of an Abuna's Chant (or whatever life-gaining instant you want to use) will be Layer 5 thanks to Boon Reflection. Then the casting of Abuna's Chant gets layers 6-8 and Surreal Memoir gets layers 9-10. Then I guess I forgot about being able to use Rings of Brighthearth on the clue tokens, so that gets layer 11, and then Press for Freedom gets layers 12-15.
I'm confused by your last edit: Why do we want to get Surreal Memoir into our yard, I thought we wanted to draw it with Clue tokens, and how would Quasiduplicate fix that?
Layer 1: Creating a token copy of Doubling Season
Layer 2: Activating Vitu-Ghazi Guildmage, and copying it for each Illusionist's Bracers
Layer 3: Resolving a Cradle of Vitality trigger puts many counters on a Runaway-Steam Kin thanks to Boon Reflection and Doubling Season
Layer 4: Gaining life triggers many Cradles of Vitality
Layer 5: I think (?) resolving a single Fiery Confluence will create many lifegain triggers because of all the creatures on the battlefield
Layers 6-7: Casting Fiery Confluence gets two layers worth of copies thanks to Precursor Golem
Layer 8: Casting from you hand gets many castings from your graveyard thanks to the Harness the Storm combo
As you can see, the Wydwen / Lobber Crew combo does not add any layers here. Since Lobber Crew just deals damage, it doesn't connect to any of the lower layers, so Wydwen / Lobber Crew are part of their own separate chain that don't lengthen the main chain.
The Press for Answers / Surreal Memoir combo is nice... if we added an instant that adds life (say Abuna's Chant) we would have:
Layers 5-6: Casting Abuna's Chant gets many copies thanks to Precursor Golem
Layer 7: Casting Abuna's Chant from your hand gets many castings from the graveyard
Layer 8: Casting Surreal Memoir gets many copies
Layer 9: Casting Surreal Memoir from your hand gets many castings from the graveyard
Layer 10: Resolving a Press for Anwers gets many Clue tokens
Layers 11-12: Casting Press for Answers gets many copies
Layer 13: Casting Press for Answers from your hand gets many castings from your graveyard
So you're up to layer 13! Not bad at all.
And yeah, you definitely want more Inexorable Tides, since they resolve one at a time, whereas Runaway Steam-Kins will be populated all at once. (They get counters one at a time from red spells, but Inexorable Tides is a teeny bit better even then because you get counters on all RSKs for each resolution rather than just one.)
I'm just going to mention one thing before I analyze the deck further. I just realized that Inexorable Tide just causes spells cast to be at Layer 3, not Layer 4. This is because Inexorable Tide puts counters on all Runaway Steam-Kins at the same time. This means they get affected by exactly the same number of Doubling Seasons, so if you have R Runaway Steam-Kins and D Doubling Seasons, Inexorable Tide resolves as a single step that adds R * 2^D counters in total to your RSKs. This is what we call a "multiplicative effect", and, while that sounds like it is good, what it means is that the layers do not stack; having lots of Doubling Seasons with one RSK would count for a layer, and having lots of RSKs with no Doubling Seasons would count for a layer, but having them both is still just a single layer, it just means that the number to the right of the arrows is bigger, but that advantage will disappear as you go up in layers, as I've tried to show in my previous analyses. So Inexorable Tide is at layer 3.
Contrast that with what happens when you cast a red spell; then, every Runaway Steam-Kin will trigger, and rather than all resolve at once, all those triggered abilities go on the stack to be resolved one at a time. So, you resolve one RSK trigger, and get a whole bunch of counters because of your number of Doubling Seasons. You can then start removing the counters to get red mana and populate, increasing the Doubling Seasons by a lot (at the Layer 2 level). Then, when you resolve the next RSK trigger, there will be a whole bunch more counters created because there are so many more Doubling Seasons in play now. So that is exactly what it means for layers to stack.
So in short, it would be better to go back to Painter's Servant, provided that that does not cause an infinite; if it does, then we will need red spells to get to Layer 4.
And, just as I'm typing this, I realize that you can have a whole bunch of Inexorable Tides, and they will all resolve one at a time. So I'm being silly again. Well, I'll leave it because the above explains why Inexorable Tide is still at Layer 4 and not Layer 5, even though we can add counters to all RSKs at once and not just one.
The problem with cards like Niv-Mizzet, Parrun is that it will mess up our spell chain, since any card can then fetch any other. To avoid this, we would have to prevent cards going back into our library - but most of our chain aside from Emergency Powers gets cards from the library. It would be nice to find some sort of card drawer that was necessarily restricted in some way - perhaps depending on combat, or the opponent's library.
Electrodominance looks very good, we can preserve the other colors besides red.
Edit: Seafloor Oracle looks quite good. Get 3 or 4 of them out and attack, draw at least 9 or 16 cards.
Edit: Ooh, I found a way to extend the chain by one. Instead of Aegis of the Heavens, the base spell will be Yawgmoth's Vile Offering which will bring back Forerunner of the Empire, then a Dinosaur, to do another three rounds worth of damaging. Then we can bring back YVO using Board the Weatherlight, and then we can fetch Board the Weatherlight with Ajani's Influence. So that should bring us to 2^^^^19!
Edit: Pfft, I'm being an idiot. We don't have Vedalken Orrery! So any sorcery that we try to fetch will be fetched exactly once, then it's stuck in our hand. So we need everything but the last spell in the chain to be instant speed.
We can still get a chain of three: Aegis of the Heavens -> Sphinx's Insight -> Emergency Powers
That results in between 2^^21 and 2^^22 damage when all is said and done. Surely we can do better!
Edit: We can extend the chain by one again. This time the base spell is Spit Flame, which we can use to damage a Polyraptor, creating all of one token copy. But that's good enough to start with. Spit Flame can be brought back to our hand when a Dragon comes into the battlefield under our control. So, the next spell up will be Yawgmoth's Vile Offering which can bring a random Dragon back onto the battlefield. YVO is a sorcery, but we can give it flash with Raff Capashen, Ship's Mage. We can destroy our Dragon using a spare copy of Spit Flame, to be brought back again with YVO. YVO then leads into Sphinx's Insight, and then Emergency Powers, as before.
Also, why do you need Masterwork of Ingenuity? You can just as easily populate Illusionist's Bracers.
Emergency Powers will work, so we can go Ajani's Influence -> Precognitive Perception -> Emergency Powers for instance.
Edit: So, I've thought about making a Standard deck a little. Like Stakfish said, Thousand-Year Storm seems like a good card to abuse; we want to set up as long an instant/sorcery chain to make it happen. Ideally, we would like to be able to create many token copies of Thousand-Year Storm, but I don't know of any way to make tokens of enchantments in Standard, and there probably isn't a way.
So I was looking at something like:
1 Forerunner of the Empire
4 Thousand-Year Storm
1 Ajani's Influence
1 Expansion // Explosion
4 Flood of Recollection
4 Vivid Revival
Obviously, we need a way to get started, a way to generate mana for all our spells, and a way to shuffle cards from our graveyard into our library. So this is just a starting analysis.
Polyraptor and Forerunner of the Empire is a well-known combo; if Forerunner is in play, and we play Polyraptor, we will generate 8 Polyraptors without any interference. More generally, if the Forerunner has X toughness, we will generate about c^X Polyraptors, where c is a little less than 2. So buffing up our Forerunner leads to exponentially more Polyraptors.
Aegis of the Heavens can pump the toughness of Forerunner of the Empire, and with four Thousand-Year Storms, we can get many copies. Casting Ajani's Influence can fetch Aegis of the Heavens many times; if the TYS count is at X, each TYS trigger gets us X copies and allows us to cast Aegis of the Heavens X times, updating the count to 2X. So four triggers will take us to 16X. Expansion // Explosion can fetch Ajani's Influence many times, and one TYS trigger will get X copies, taking X to X*16^X. So four triggers will take X to more than 16^16^16^16^X. Finally, each copy of Flood of Recollection or Vivid Revival can fetch Expansion // Explosion many times, with a single TYS trigger taking X to more than 2^^(4X). With 8 spells and 4 TYS's we get 32 triggers in all, so we will finally reach more than 2^^^35 castings of Aegis of the Heavens. We then get an extra exponent when we count the Polyraptors, but of course that is unnoticeable.
This is just a start, but it looks decent so far.
Edit: Just realized that targeted spells don't play nicely with lots of spell copies entering the stack at the same time. So yeah, we want to replace Flood of Recollection and Vivid Revival with nontargeted spells.
With the Harness the Storm combo, Spitting Image would get to Layer 4. But even that would make basically no difference. Having two chains at level 4 still takes X to roughly X^^^^X, except that with this both the number of Runaway Steam-Kins and the number of Harness the Storms count. But, it's not even worth it to populate both Runaway Steam-Kin and Harness the Storm; it's much better to populate another Doubling Season, then populate one or the other of RSK and HtS, since that gives you exponentially more. So in fact even X is basically the same here.
Basically, it's really inefficient to have multiple chains up to a particular layer, since you basically get the same benefit with just one, so if you create multiples you wind up using more cards and cutting off more avenues for continuation. So yeah, you should pick one of the two Layer 4 chains and just set up that one.
Layer 1: Creating a token creates exponentially more thanks to Doubling Season
Layer 2: Activating the Populate ability gets X copies with X Rowan Kenriths
Layer 3: Resoliving a Runaway Steam-Kin triggers gets many counters thanks to Doubling Season
Layer 4: Casting a spell triggers many Runaway Steam-Kins
So if we cast Spitting Image 20 times, along with around 40 other spells once we get the full setup going, we will get to about 2^^^^^62. We can do much better if we set up a spell chain where each spell can cause the previous spell to be cast many times. So for example:
Layers 5-7: Playing Aether Mutation gets many copies via Harness the Storm and Precursor Golem, and can return a creature back to our hand many times.
Layers 8-9: Playing Revive will allow us to fetch Aether Mutation many times.
Layers 10-11: Playing Spell Snare can counter the original Revive and allow it to fetch itself many times.
Layers 12-13: Playing Call to Mind can fetch spell Snare many times.
Layers 14-15: Playing any of the many spells that can exile themselves to fetch cards from the graveyard can fetch Call to Mind many times.
Just throwing out some ideas.
Edit: I thought of going back to Splinter Twin to get rid of Mimic Vat in the Soul Foundry deck, and possibly to fix the Chalice of the Void deck, but unfortunately it looks like we can create a hasted token of Goblin Dark-Dwellers, and then keep creating more and more copies.
Edit: Thinking some more about the Chalice of the Void deck, I have come up with an idea: What about using Allay instead of Psychatog in order to bounce as well as destroy enchantments, then add Bident of Thassa so we have an artifact that we can repeatedly bounce using all our mana. Then we go to Golem Foundry, where we can use all our artifact casts to create Golem artifact tokens, then have Disenchant, copied using Precursor Golem. As far as I know, Disenchant did not go infinite in our decks, it just wasn't good enough to recreate a stage. Does that work? Perhaps the main problem is that it requires an additional four cards, although Precursor Golem is also useful for the ending layers. So I think that would reduce the deck down to six stages, assuming nothing else changes. Still, it's good to have as many decks fixed as possible, since we don't know what might still go wrong.