Right, I'm saying we would need to put life above mana - so no having Banefire into a lifelinked Sun-Crowned Hunters. I thought that was your intention.
It looks like Bedeck / Bedazzle shows some promise, we could have a nonbasic land that could tap for white mana, along with regular Mountains. Then Gaea's Blessing could draw Bedeck // Bedazzle, and Bedazzle could destroy and recycle the nonbasic land that could tap for a lot of white mana. So we could have:
tap nonbasic white mana land - layer 1
cast Bedazzle - 2
cast Ionize - 3
resolve Enrage trigger - 4
cast Banefire - 5
Tap Mountain - 6
Teferis - 7
So that would get us to 3 -> 3 -> (3 -> 3 -> X -> 7) -> 7 by the end, an improvement of two layers. The problem is the same as what we had before - we can use Bedazzle to generate a lot of white mana, and then we can spend some of the white mana to sacrifice a locket a redraw Bedazzle. If Core 2020 comes out with better mana rocks, we could add two layers.
Edit: Okay, I've found a small improvement. Take away 5 Mountains, and add four of The First Eruption, and one Guildpact Informant. We put all four Eruptions on the battlefield, and then attack with 19 hasted Guildpact Informants. This will allow us to proliferate 19 times, getting to the third step of an eruption 4*9 = 36 times. This allows us to sacrifice and recycle 36 Mountains, making up for the 5 Mountain loss and then some. We also add 4*2*10 = 80 red mana, and 36 times we can deal a bunch of damage to a bunch of Sun-Crowned Hunters, although this is comparatively minor compared to getting more Mountain taps. So this boosts our Mountain taps by 31, taking us to 3 -> 3 -> (3 -> 3 -> 56 -> 5) -> 5.
Edit: Even better: Take away 3 Mountains, Grand Warlord Radha, and Grow from the Ashes. Add three more copies of The Mending of Dominaria, one copy of Fall of the Thran and one copy of Guildpact Informant. Adding Fall of the Thran gives us the ability to destroy our lands at any time, but removing Grow from the Ashes prevents us from putting them on the battlefield again. Our only way is to put them into the graveyard by milling or destroying them, then proliferating to return them from the graveyard to the battlefield, using either The Mending of Dominaria or Fall of the Thran.
So, after putting our lands in our graveyard, and putting all four The Mending of Dominarias and Fall of the Thran on the battlefield, we attack with 19 Guildpact Informants, and after damage is dealt we put 19 proliferate triggers on the stack. The first trigger resolves, moving Fall of the Thran to the second step and returning two Mountains. We can't really take advantage of them yet, since we don't want to destroy our The Mending of Dominarias yet. We resolve the second proliferate, getting all 5 enchantments to the third step. We resolve the Fall of the Thran trigger first, returning two more Mountains. Now we can go ahead and tap the Mountains one by one, each time with increasing Nikyas. This will have the side effect of destroying our enchantments, and we can bring Fall of the Thran back at least once to destroy our four Mountains after we are done tapping them. Then we resolve the TMoD triggers one at a time, each time bringing back all 21 of our Mountains, which we can destroy with a Fall of the Thran after tapping them all. So with two proliferate triggers we get 4 + 4*21 = 88 Mountain returns. So with 19 proliferate triggers, we get 9 * 88 + 2 = 794 Mountain returns, allowing us to increase our final damage to more than 3 -> 3 -> (3 -> 3 -> 794 -> 5) -> 5, assuming it all works as planned.
Edit: I've found another improvement. Remove Grand Warlord Radha, Grow from the Ashes, and as many Mountains as we need to (probably 23). Add Fall of the Thran, Guildpact Informant, Dowsing Dagger, Gideon, the Oathsworn, and Role Reversal. Now, we can cast Dowsing Dagger, forcing the opponent to create two creatures, and then we can cast Role Reversal. The first Role Reversal can switch one of the opponent's creatures with Gideon, the Oathsworn, and the second Role Reversal can switch Gideon with Teferi, Hero of Dominaria. Then, during the combat phase, we can create as many Guildpact Informants as we can, and then have them all attack Teferi, allowing them all to deal combat damage without killing the opponent. This gets us a proliferate trigger for every Guildpact Informant we can create; with 2X triggers, we can deal more than 3 -> 3 -> (3 -> 3 -> X -> 5) -> 5 damage.
Since we can now only bring in Mountains through proliferating, it's not really all that great to have more than one. That leaves us 20 cards which we can use to increase that starting X as much as possible. Unfortunately, getting mana from Grand Warlord Radha, or by transforming Dowsing Dagger into Lost Vale, only happens after we declare attacking creatures, at which point it is too late to increase the number of Guildpact Informants. Adding four Smothering Tithes and one Unmoored Ego allows us to generate 212 red mana, so that seems like a good choice. After that, I don't know any other limited way to gain red mana. We can increase our life up to 320 with Revival // Revenge. After that, I don't know - we can always add 14 more cheap instants/sorceries that we can use to increase our TYS count on a regular basis.
Anyway, with the 212 red mana we can increase X to more than 3 -> 3 -> 213 -> 4, allowing us to increase our final damage to more than 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 213 -> 4) -> 5) -> 5.
Does anyone know other ways to gain a limited amount of red mana?
Edit: Oh wait, if we're going to have a lot of Guildpact Informants, it's better to go back to the version with The First Eruption. So take the last deck above, remove Fall of the Thran, and add The First Eruption and Grow from the Ashes. That leaves room for 20 Mountains, and with Grow from the Ashes, we can put them into play and create more than 3 -> 3 -> 21 -> 5 Guildpact Informants. We can then do the same as above to gain more than 3 -> 3 -> 21 -> 5 proliferate triggers, each of which will allow us to sacrifice a Mountain, so that we can recycle it and tap it again. That takes us to more than 3 -> 3 -> (3 -> 3 -> 21 -> 5) -> 5 red mana, which we can use to gain that many Teferi activations, and generate a final damage of more than 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 21 -> 5) -> 5) -> 5.
Hmm, Spreading Rot won't go infinite so long as we put life above (i.e. more scarce than) mana, which I think would be the point of having Spreading Rot in the first place. Same with shocklands. So the question is, can we get more layers by putting mana before life?
So, we have to figure out how we transition from tapping lands to generating stages. The obvious way is to have lands that tap for white mana. So, with a lot of Nikyas in play, we can tap a land to generate a lot of white mana, then use that white mana to execute a stage. Then we can recycle the land at the cost of 2 life. So each 2 life gets us one stage. The problem is, that is the same as what we got with Ionize, and with Ionize we didn't have to give up lands for further up the chain. So this is strictly worse.
There may be something we can make of switching up life and lands, but we have to get more out of lands than tapping for white mana.
Okay, so the Standard deck seems to be progressing along well, while the Vintage deck has stalled for a little while. Anyway, I figured now would be a good time to start thinking about creating a Modern deck.
Then, we come to what appears to be the most damaging loss: both Psychic Battle and Grip of Chaos are gone. So the normal way of processing our Vintage stage is no longer available. The best thing I can think of is to use Rings of Brighthearth in their place. This means that we can't use Mimic Vat for stages, since we could use Rings of Brighthearth on it and get multiple creature tokens per activation. So no more creature stages, back to mana stages.
Now, the champion of the mana stages previously was Chrome Mox - but it's banned in Modern! Oh well.
So, we can start perhaps by going all the way back to the very early days of our stage decks, using creatures that can discarded to create mana. These include landcyclers, the Spirit Guides, and Skirge Familiar. Some of the landcyclers are gone, as are Elvish Spirit Guide and Skirge Familliar. So it might be tough to get a full 6/7 mana stages. (and a life stage maybe?)
Of course, we could try to create a stage some other way, perhaps using the ideas from the recent Standard decks.
@FortyTwo: I just noticed your edit concerning adding Angrath's Rampage. That would be killer if we could do so, so it might be worth investigating again. If Angrath's Ramage is the spell that can get rid of lands, then maybe it should be the "higher" stage (higher of course meaning lower on the stack, as usual). We can eliminate artifacts with Cleansing Nova, so perhaps sacrificing planeswalkers could be the other thing that AR does. I think we had an idea of using Interplanar Beacon, so that recycling planeswalkers could give us life. I think the issue we had was how to recycle the planeswalker though. Anyway, I think it's worth thinking about, let's throw out any ideas that anyone comes up with!
Edit: Wait, after a Purify Nova, we won't have Arcane Adaptation on the battlefield, so I don't know how we can retrieve it with Tragic Poet. So maybe Regisaur Alpha doesn't work.
@J_kibbs: Okay, I guess I should explain how much a cheaper Teferi would gain us. In the current deck, we can get our TYS count to more than 3 -> 3 -> 1 -> 5 before we generate any red mana, but it looks like we don't reach 3 -> 3 -> 2 -> 5. We can then create a bunch of Grand Warlord Radhas and attack with everything, generating a lot more than 3 -> 3 -> 1 -> 5 red mana, by spending the red mana for Banefires, we can increase our count to more than 3 -> 3 -> 2 -> 5, but less than 3 -> 3 -> 3 -> 5. Similarly, each Mountain we tap with a lot of Nikyas in play can increase the third number in the estimation by one. So after 23 Mountains are tapped, our TYS count goes to between 3 -> 3 -> 25 -> 5 and 3 -> 3 -> 26 -> 5. This allows us to cast Quasiduplicate and create more than 3 -> 3 -> 25 -> 5 Nikyas, so our 24th Mountain generates more than 3 -> 3 -> 25 -> 5 red mana. We can then turn that red mana into Teferi activations on a one-for-one basis, so we get more than 3 -> 3 -> 25 -> 5 Teferi activations in the end. We can also use those Banefires to increase our TYS count to more than 3 -> 3 -> 26 -> 5, but that is insignificant; it's the Teferi activations that will do the heavy lifting. So in the end step, we can untap Mountains more than 3 -> 3 -> 25 -> 5 times, each time allowing us to tap the Mountain for red mana and increase the third number by one, so we wind up with a TYS count of more than 3 -> 3 -> (3 -> 3 -> 25 -> 5) -> 5 in the end, allowing us to generate that much green mana and Banefire for more than that much damage. (Remembering to save the last red mana for that Banefire, of course)
Now, if we can get many Teferi activations for a red mana, then we could use that final Mountain to increase our TYS count, until we could use the final red mana to generate more than 3 -> 3 -> 26 -> 5 Teferi activations. So the 25 would become a 26. Of course, if we had to spend a card slot to get a cheap Teferi activation, then we would have to give up a Mountain, so the 26 would go back down to 25. So for example, it looks like a second Teferi does not go infinite (we can use one Teferi to draw the other, and play it to put the original Teferi in the graveyard. But, we would still need to spend resources to move that Teferi from the graveyard to the library, so it doesn't look like we could do that forever.), and allows us to get many Teferi activations cheaply, so we would also get 3 -> 3 -> (3 -> 3 -> 25 -> 5) -> 5 with a second Teferi. It's hard to pick between the two, but I suspect that 24 Mountain version might do a touch more damage - we can fully run the combo all the way through the 23rd Mountain in that version, whereas in the 23 Mountain version, our number of Teferi activations would be based on the TYS count _prior_ to the very last Gaea's Blessing that we cast while spending the 23rd Mountain's mana. So that is smaller by an absolutely infitesimal amount, but it does look like the 24 Mountain version ekes out ahead.
Also, I agree with FortyTwo that we want to delay the tapping of each Mountain as long as possible, to generate the largest possible TYS count and create the most Nikyas. Grand Warlord Radha should go first; once you get that early lead, the difference will keep increasing as we continue the combo, so getting the advantage as early as possible is best.
Edit: Eh, my reasoning for going with 24 Mountains and one Teferi is not compelling. That minute difference will only matter if everything else is the same up to that point; if there is even a little bit of gain we can get earlier by having two Teferis, that should be enough to tip the scales in that direction.
Unfortunately, at first glance it looks like we get only one Teferi activation per Banefire cast; if you see a way to get more, that would be great. We can get Teferi back on the battlefield without too much expenditure, even if we have to draw it, but destroying it costs us an Enrage trigger. So we have to cast Banefire to get it off the battlefield, but we can't use it and destroy it again while there are copies of Banefire on the stack, since we have to wait until all the Banefires are gone until we can actually activate it. So one activation per Banefire cast.
You don't really need to jump through hoops, you just need one other card. (like Finale of Devastation or many other possibilities) If you look at the comments of that twitter post, there are a few references to it being a three card combo (Ghalta and Benalish Marshal are mentioned.).
I actually did ask this question in the Rules forum, and got the answer that it doesn't get a +1/+1 counter if you don't copy anything. There is the same answer here as well.
Vermorax, thanks for sharing and making a nice writeup!
Make a Wish and Twincast go infinite. With at least one Make a Wish on the stack, we can cast Twincast, having all the copies copy Make a Wish. The copies of Twincast can fetch whatever, then when we resolve the original, it makes a copy of Make a Wish, then goes to the graveyard. Then we resolve the Make a Wish copy, and we can fetch Twincast with it, and cast it again.
Without Twincast, it looks like:
Each casting of Clone Legion takes X to more than 2^^X
Each casting of Make a Wish takes X to more than 2^^^X
Each casting of a self-exiling fetch card takes X to more than 2^^^^X
So with X self-exiling fetch cards, we produce more than 2^^^^^(X+2) birds in the end.
Yeah, we can't target Gideon with Sun-Crowned Hunters, since it doesn't actually deal the damage. So, we can either reshuffle and redraw Teferi each time after destroying it (we can save four white mana in order to draw by sacrificing a locket), or we can use Spark Double to to copy it, and then reshuffle and refetch it by casting Adventurous Impulse from the library. Neither is particularly irksome in terms of resource expenditure.
Fortunately, the +1/+1 clause in Spark Double is tied to the initial clause, which is "You may have Spark Double enter the battlefield as a copy of a creature or planeswalker you control". So it's not just entering the battlefield, it's entering the battlefield as a copy of something. Hence, if it copies nothing, it doesn't get a +1/+1 counter and goes to the graveyard.
Vivien's Grizzly is an interesting card, but with our current setup it goes infinite, since we will be able to reshuffle at the cost of a few mana (which allows us to recast Gift of Growth for more mana, among other things).
Ah, good point. As you say, we can use the previous stage to be able to cast a Wrath Nova. When we execute a stage, we start with a stage with X layers, then X-1 layers, then X-2 layers, etc. So when we are finally down to just two white mana in our mana pool, we stop the combo there if we are out of life. Then we cast a Quasiduplicate to gain as many Angrath's Marauders as possible, resolve a Banefire on Sun-Crowned Hunters to deal a ton of damage to Teferi and gain us a lot of life, then we can cast a Wrath Nova from the library, followed by Gaea's Blessing and Ionize (which we use a draw to bring to our hand).
I am certainly aware of the difficulty that most people will have trying to understand both the deck mechanics and the large number systems we are using in these challenges. I have tried my best at various points in the other thread to try to explain various points on what is going on in our Vintage deck, and both Iijil and Stakfish have made explanatory posts as well. But, I don't really know how successful those posts were at clarifying things for other people.
The OP does have some links already for some of the explanations that we have written up: there is a link to SadisticMystic's previous write-ups of the layer decks; a link to my write-up of the megastage deck, which is currently unfinished, but has explanations for how the stage and the hyperstage operate: a link to my post describing the operation of the original stage combo at the time I discovered it; a link to Iijil describing how we could go about extending a stage to a hyperstage; and a link to a post by Stakfish giving an overview of how the most current decks work, going through card by card. There were a few other posts in the other thread that were explanatory attempts, but I would have to dig through the thread to find them. A link to the Googology Wiki seems like a good idea.
A newcomer thread might be helpful, but I'm worried that we would be getting into the territory of too many threads. We are sharing a public forum with many people who want to discuss Magic in general, so having too many threads devoted to maximum number challenges might be annoying to some others who aren't interested in this. If we were to move to a server hosted by Stakfish, we could make as many threads as we want in our own forum, but I for one would prefer to move to a more popular forum, so that others might take a glance, like what they see, and join in on the challenges. So in the latter case, we won't have our own subforum, and maybe should be careful how many threads we create.
Concerning the decklist, previously we made a slight improvement by switching out one Mountain for a Grand Warlord Radha. It's not much, but still an improvement.
It looks like Bedeck / Bedazzle shows some promise, we could have a nonbasic land that could tap for white mana, along with regular Mountains. Then Gaea's Blessing could draw Bedeck // Bedazzle, and Bedazzle could destroy and recycle the nonbasic land that could tap for a lot of white mana. So we could have:
tap nonbasic white mana land - layer 1
cast Bedazzle - 2
cast Ionize - 3
resolve Enrage trigger - 4
cast Banefire - 5
Tap Mountain - 6
Teferis - 7
So that would get us to 3 -> 3 -> (3 -> 3 -> X -> 7) -> 7 by the end, an improvement of two layers. The problem is the same as what we had before - we can use Bedazzle to generate a lot of white mana, and then we can spend some of the white mana to sacrifice a locket a redraw Bedazzle. If Core 2020 comes out with better mana rocks, we could add two layers.
Edit: Okay, I've found a small improvement. Take away 5 Mountains, and add four of The First Eruption, and one Guildpact Informant. We put all four Eruptions on the battlefield, and then attack with 19 hasted Guildpact Informants. This will allow us to proliferate 19 times, getting to the third step of an eruption 4*9 = 36 times. This allows us to sacrifice and recycle 36 Mountains, making up for the 5 Mountain loss and then some. We also add 4*2*10 = 80 red mana, and 36 times we can deal a bunch of damage to a bunch of Sun-Crowned Hunters, although this is comparatively minor compared to getting more Mountain taps. So this boosts our Mountain taps by 31, taking us to 3 -> 3 -> (3 -> 3 -> 56 -> 5) -> 5.
Edit: Even better: Take away 3 Mountains, Grand Warlord Radha, and Grow from the Ashes. Add three more copies of The Mending of Dominaria, one copy of Fall of the Thran and one copy of Guildpact Informant. Adding Fall of the Thran gives us the ability to destroy our lands at any time, but removing Grow from the Ashes prevents us from putting them on the battlefield again. Our only way is to put them into the graveyard by milling or destroying them, then proliferating to return them from the graveyard to the battlefield, using either The Mending of Dominaria or Fall of the Thran.
So, after putting our lands in our graveyard, and putting all four The Mending of Dominarias and Fall of the Thran on the battlefield, we attack with 19 Guildpact Informants, and after damage is dealt we put 19 proliferate triggers on the stack. The first trigger resolves, moving Fall of the Thran to the second step and returning two Mountains. We can't really take advantage of them yet, since we don't want to destroy our The Mending of Dominarias yet. We resolve the second proliferate, getting all 5 enchantments to the third step. We resolve the Fall of the Thran trigger first, returning two more Mountains. Now we can go ahead and tap the Mountains one by one, each time with increasing Nikyas. This will have the side effect of destroying our enchantments, and we can bring Fall of the Thran back at least once to destroy our four Mountains after we are done tapping them. Then we resolve the TMoD triggers one at a time, each time bringing back all 21 of our Mountains, which we can destroy with a Fall of the Thran after tapping them all. So with two proliferate triggers we get 4 + 4*21 = 88 Mountain returns. So with 19 proliferate triggers, we get 9 * 88 + 2 = 794 Mountain returns, allowing us to increase our final damage to more than 3 -> 3 -> (3 -> 3 -> 794 -> 5) -> 5, assuming it all works as planned.
Edit: I've found another improvement. Remove Grand Warlord Radha, Grow from the Ashes, and as many Mountains as we need to (probably 23). Add Fall of the Thran, Guildpact Informant, Dowsing Dagger, Gideon, the Oathsworn, and Role Reversal. Now, we can cast Dowsing Dagger, forcing the opponent to create two creatures, and then we can cast Role Reversal. The first Role Reversal can switch one of the opponent's creatures with Gideon, the Oathsworn, and the second Role Reversal can switch Gideon with Teferi, Hero of Dominaria. Then, during the combat phase, we can create as many Guildpact Informants as we can, and then have them all attack Teferi, allowing them all to deal combat damage without killing the opponent. This gets us a proliferate trigger for every Guildpact Informant we can create; with 2X triggers, we can deal more than 3 -> 3 -> (3 -> 3 -> X -> 5) -> 5 damage.
Since we can now only bring in Mountains through proliferating, it's not really all that great to have more than one. That leaves us 20 cards which we can use to increase that starting X as much as possible. Unfortunately, getting mana from Grand Warlord Radha, or by transforming Dowsing Dagger into Lost Vale, only happens after we declare attacking creatures, at which point it is too late to increase the number of Guildpact Informants. Adding four Smothering Tithes and one Unmoored Ego allows us to generate 212 red mana, so that seems like a good choice. After that, I don't know any other limited way to gain red mana. We can increase our life up to 320 with Revival // Revenge. After that, I don't know - we can always add 14 more cheap instants/sorceries that we can use to increase our TYS count on a regular basis.
Anyway, with the 212 red mana we can increase X to more than 3 -> 3 -> 213 -> 4, allowing us to increase our final damage to more than 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 213 -> 4) -> 5) -> 5.
Does anyone know other ways to gain a limited amount of red mana?
Edit: Oh wait, if we're going to have a lot of Guildpact Informants, it's better to go back to the version with The First Eruption. So take the last deck above, remove Fall of the Thran, and add The First Eruption and Grow from the Ashes. That leaves room for 20 Mountains, and with Grow from the Ashes, we can put them into play and create more than 3 -> 3 -> 21 -> 5 Guildpact Informants. We can then do the same as above to gain more than 3 -> 3 -> 21 -> 5 proliferate triggers, each of which will allow us to sacrifice a Mountain, so that we can recycle it and tap it again. That takes us to more than 3 -> 3 -> (3 -> 3 -> 21 -> 5) -> 5 red mana, which we can use to gain that many Teferi activations, and generate a final damage of more than 3 -> 3 -> (3 -> 3 -> (3 -> 3 -> 21 -> 5) -> 5) -> 5.
So, we have to figure out how we transition from tapping lands to generating stages. The obvious way is to have lands that tap for white mana. So, with a lot of Nikyas in play, we can tap a land to generate a lot of white mana, then use that white mana to execute a stage. Then we can recycle the land at the cost of 2 life. So each 2 life gets us one stage. The problem is, that is the same as what we got with Ionize, and with Ionize we didn't have to give up lands for further up the chain. So this is strictly worse.
There may be something we can make of switching up life and lands, but we have to get more out of lands than tapping for white mana.
The first thing that I noticed among card legalities is the absence of Opalescence. At first I thought that this would be extremely damaging, but then I realized that Starfield of Nyx could be a suitable substitite. Among other cards, Bloodbond March, Cowardice, Omniscience, and Leyline of Anticipation / Vedalken Orrery are all legal. But Cephalid Shrine is not - does anyone know a good substitute for Cephalid Shrine? Declaration of Naught and Draining Whelk are possibilities, although they require blue mana and recycling a creature respectively.
Then, we come to what appears to be the most damaging loss: both Psychic Battle and Grip of Chaos are gone. So the normal way of processing our Vintage stage is no longer available. The best thing I can think of is to use Rings of Brighthearth in their place. This means that we can't use Mimic Vat for stages, since we could use Rings of Brighthearth on it and get multiple creature tokens per activation. So no more creature stages, back to mana stages.
Now, the champion of the mana stages previously was Chrome Mox - but it's banned in Modern! Oh well.
So, we can start perhaps by going all the way back to the very early days of our stage decks, using creatures that can discarded to create mana. These include landcyclers, the Spirit Guides, and Skirge Familiar. Some of the landcyclers are gone, as are Elvish Spirit Guide and Skirge Familliar. So it might be tough to get a full 6/7 mana stages. (and a life stage maybe?)
Of course, we could try to create a stage some other way, perhaps using the ideas from the recent Standard decks.
Regisaur Alpha might be better, but we have to check how we go about changing around the Arcane Adaptations. We need to switch to Wizard each time we cast Excavation Elephant, then switch to Dinosaur so that we can tap our Tragic Poets, Marwyn, and Gyre Engineers.
Edit: Wait, after a Purify Nova, we won't have Arcane Adaptation on the battlefield, so I don't know how we can retrieve it with Tragic Poet. So maybe Regisaur Alpha doesn't work.
@J_kibbs: Okay, I guess I should explain how much a cheaper Teferi would gain us. In the current deck, we can get our TYS count to more than 3 -> 3 -> 1 -> 5 before we generate any red mana, but it looks like we don't reach 3 -> 3 -> 2 -> 5. We can then create a bunch of Grand Warlord Radhas and attack with everything, generating a lot more than 3 -> 3 -> 1 -> 5 red mana, by spending the red mana for Banefires, we can increase our count to more than 3 -> 3 -> 2 -> 5, but less than 3 -> 3 -> 3 -> 5. Similarly, each Mountain we tap with a lot of Nikyas in play can increase the third number in the estimation by one. So after 23 Mountains are tapped, our TYS count goes to between 3 -> 3 -> 25 -> 5 and 3 -> 3 -> 26 -> 5. This allows us to cast Quasiduplicate and create more than 3 -> 3 -> 25 -> 5 Nikyas, so our 24th Mountain generates more than 3 -> 3 -> 25 -> 5 red mana. We can then turn that red mana into Teferi activations on a one-for-one basis, so we get more than 3 -> 3 -> 25 -> 5 Teferi activations in the end. We can also use those Banefires to increase our TYS count to more than 3 -> 3 -> 26 -> 5, but that is insignificant; it's the Teferi activations that will do the heavy lifting. So in the end step, we can untap Mountains more than 3 -> 3 -> 25 -> 5 times, each time allowing us to tap the Mountain for red mana and increase the third number by one, so we wind up with a TYS count of more than 3 -> 3 -> (3 -> 3 -> 25 -> 5) -> 5 in the end, allowing us to generate that much green mana and Banefire for more than that much damage. (Remembering to save the last red mana for that Banefire, of course)
Now, if we can get many Teferi activations for a red mana, then we could use that final Mountain to increase our TYS count, until we could use the final red mana to generate more than 3 -> 3 -> 26 -> 5 Teferi activations. So the 25 would become a 26. Of course, if we had to spend a card slot to get a cheap Teferi activation, then we would have to give up a Mountain, so the 26 would go back down to 25. So for example, it looks like a second Teferi does not go infinite (we can use one Teferi to draw the other, and play it to put the original Teferi in the graveyard. But, we would still need to spend resources to move that Teferi from the graveyard to the library, so it doesn't look like we could do that forever.), and allows us to get many Teferi activations cheaply, so we would also get 3 -> 3 -> (3 -> 3 -> 25 -> 5) -> 5 with a second Teferi. It's hard to pick between the two, but I suspect that 24 Mountain version might do a touch more damage - we can fully run the combo all the way through the 23rd Mountain in that version, whereas in the 23 Mountain version, our number of Teferi activations would be based on the TYS count _prior_ to the very last Gaea's Blessing that we cast while spending the 23rd Mountain's mana. So that is smaller by an absolutely infitesimal amount, but it does look like the 24 Mountain version ekes out ahead.
Also, I agree with FortyTwo that we want to delay the tapping of each Mountain as long as possible, to generate the largest possible TYS count and create the most Nikyas. Grand Warlord Radha should go first; once you get that early lead, the difference will keep increasing as we continue the combo, so getting the advantage as early as possible is best.
Edit: Eh, my reasoning for going with 24 Mountains and one Teferi is not compelling. That minute difference will only matter if everything else is the same up to that point; if there is even a little bit of gain we can get earlier by having two Teferis, that should be enough to tip the scales in that direction.
I actually did ask this question in the Rules forum, and got the answer that it doesn't get a +1/+1 counter if you don't copy anything. There is the same answer here as well.
Vivien's Grizzly allows us to put Gaea's Blessing in our graveyard, for an easy reshuffle.
Make a Wish and Twincast go infinite. With at least one Make a Wish on the stack, we can cast Twincast, having all the copies copy Make a Wish. The copies of Twincast can fetch whatever, then when we resolve the original, it makes a copy of Make a Wish, then goes to the graveyard. Then we resolve the Make a Wish copy, and we can fetch Twincast with it, and cast it again.
Without Twincast, it looks like:
Each casting of Clone Legion takes X to more than 2^^X
Each casting of Make a Wish takes X to more than 2^^^X
Each casting of a self-exiling fetch card takes X to more than 2^^^^X
So with X self-exiling fetch cards, we produce more than 2^^^^^(X+2) birds in the end.
Fortunately, the +1/+1 clause in Spark Double is tied to the initial clause, which is "You may have Spark Double enter the battlefield as a copy of a creature or planeswalker you control". So it's not just entering the battlefield, it's entering the battlefield as a copy of something. Hence, if it copies nothing, it doesn't get a +1/+1 counter and goes to the graveyard.
Vivien's Grizzly is an interesting card, but with our current setup it goes infinite, since we will be able to reshuffle at the cost of a few mana (which allows us to recast Gift of Growth for more mana, among other things).
The OP does have some links already for some of the explanations that we have written up: there is a link to SadisticMystic's previous write-ups of the layer decks; a link to my write-up of the megastage deck, which is currently unfinished, but has explanations for how the stage and the hyperstage operate: a link to my post describing the operation of the original stage combo at the time I discovered it; a link to Iijil describing how we could go about extending a stage to a hyperstage; and a link to a post by Stakfish giving an overview of how the most current decks work, going through card by card. There were a few other posts in the other thread that were explanatory attempts, but I would have to dig through the thread to find them. A link to the Googology Wiki seems like a good idea.
A newcomer thread might be helpful, but I'm worried that we would be getting into the territory of too many threads. We are sharing a public forum with many people who want to discuss Magic in general, so having too many threads devoted to maximum number challenges might be annoying to some others who aren't interested in this. If we were to move to a server hosted by Stakfish, we could make as many threads as we want in our own forum, but I for one would prefer to move to a more popular forum, so that others might take a glance, like what they see, and join in on the challenges. So in the latter case, we won't have our own subforum, and maybe should be careful how many threads we create.
Concerning the decklist, previously we made a slight improvement by switching out one Mountain for a Grand Warlord Radha. It's not much, but still an improvement.