Hmm yeah, with Mystic Sanctuary we can't ever use Saw in Half on anything other than Astral Dragon, so we'll only have one copy of other creatures. Heliod, Sun-Crowned can be a noncreature permanent at the start, but once we start making many of them, that will stop. So we can't keep growing the number. And with limited triggers, Heliod and Archangel of Thune won't be worth a layer.
Druids' Repository plus Herald of the Sun looks good though. Oh, but there's an issue: we want all the cards in play besides Saw in Half when we start drawing, but that's 9 cards, and we only draw 1 from the initial Astral Dragon. We'll draw more when we cast Herald of the Sun, but we won't be able to play Mystic Sanctuary or Druids' Repository while those draws are on the stack. We need Mystic Sanctuary, so that would just leave us with one Druids' Repository... Oh, but that's good enough, since one will still trigger for every creature that attacks. Good.
So the start is the same as the one that plopfill explained; we get 5 draws from casting Herald of the Sun, we use 3 to draw and cast Saw in Half, getting up to 2^^^5, and then we use the last two to draw Saw in Half and Druids' Repository. Go into combat with everything, Druids' Repository triggers more than 2^^^5 times, and we get a layer for all the counters we get for each DR trigger, and a layer for having 2^^^5 of them. So a total of F_{w+2}(2^^^5).
It would be nice to turn lifegain into many triggers of adding +1/+1 counters - then we could maybe get F_{w+3} with 11 cards, by going Druids' Repository to gain mana many times, pay mana to gain life many times, then either pay life to add +1/+1 counters, or get +1/+1 counters from triggers. But, I don't know what to use for the last two pieces.
How about Rings of Brighthearth? Each +1/+1 counter activation gets a new set of Rings triggers to multiply it, and then when it finishes we can make more Rings. (Mystic Sanctuary means the mana cost is trivial.)
Now that I think about it, old tech would probably work decently well here. If we add Dual Nature, along with a creature that adds +1/+1 on ETB or activation, then that should also get F_{w+2} with 10 cards, but probably a bigger input due to all the draws we get with Dual Nature. Then, a second creature that bounces the first on activation or ETB could get us to F_{w+4} with 11 cards. A third creature would get F_{w+6}, or if the first two creatures were activated abilities, we could add Rings of Brighthearth as well for F_{w+6}. Then three creatures with Rings of Brightheart could get F_{w+9}, and so on.
Edit: Oh, that old tech relied on Vedalken Orrery / Leyline of Anticipation of course. We could still get something with flickering, or with adding Leyline once we have enough cards in the deck.
I have now realised that once the power of an Astral Dragon is increased to 7+, after using Saw in Half on it, one can interleave the "draw a card" and "create copies of Mystic Sanctuary" triggers, which makes it possible to cast Saw in Half on other creatures without losing it.
Ah, okay. That kills the ETB/activation possibilities then. But it helps the versions where we trigger through something else, like combat. Patron of the Kitsune / Archangel of Thune should work now then; there's very little difference between the two versions, it's almost a tie. Better than both would be using two creatures with power 4 or more, I would imagine.
Play Saw in Half on Astral Dragon, making 4 dragons and making the Doubling Season count go 1 > 5 > 37 > 2^37, then use the last to make 2^2^37 Ascendancies. Mystic Sanctuary puts back Saw. Hunted Troll gets 2^2^37 draw triggers, each time drawing Saw, playing it on Astral Dragon for more Doubling Seasons and one more round of Mystic Sanctuaries. At the end, make another round of Ascendancies and use the last Saw on Hunted Troll, making more 4-power tokens and repeating the process.
At the end of that, make another round of Ascendancies and use the last Saw on Cadric. Play Toralf, God of Fury, triggering the Ascendancies and the Cadrics, paying for the Cadric triggers with Mystic Sanctuary mana. Repeat the process for every Cadric trigger, finishing by making more Cadrics again. Then play Ziatora, the Incinerator and do the same thing.
Use the last draw to pick up Rite of Passage and play it. Go to end step, let the Ziatoras trigger, and throw something unimportant at one of the opponent's tokens. Run the Toralf combo, finishing by having the last point of damage hit one of our Astral Dragons to trigger Rite of Passage, putting counters on it. When that happens, run the Saw in Half loop on it, finishing by making more Rite of Passage for more counter triggers next time, as well as getting more copies of Toralf and Hunter Troll. At the end of a Toralf round, use the last counter trigger to make the next Ziatora hit even bigger. Finish by having the last Ziatora go for our opponent's face.
Final damage should be a function of the number of Ziatora triggers, although I'm not sure how to modify the f{w} part for this.
So four of the draws can draw Saw in Half, and one can draw Hunted Troll. The first draw and cast gets us 1 > 5 > 69 > 2^70 Doubling Seasons, then we copy Mystic Sanctuary to put Saw in Half on top of the library. The second Saw in Half creates more than 2^2^70 Astral Dragons, getting us to 2^^(2^2^70) >> 2^^^4 Doubling Seasons, and the next two Saw in Halfs get us to more than 2^^^6 Doubling Seasons and Temur Ascendancies.
We get more than 2^^^6 draws off of Hunted Troll, getting us to 2^^^2^^^6 Doubling Seasons, then the last Saw on Hunted Troll creates 2^^^2^^^6 power 4 Hunted Trolls, each triggering Ascendancies, getting us to 2^^^2^^^2^^^6.
We make a bunch of Cadrics and play Toralf, God of Fury. Each Cadric trigger gets us a bunch of Ascendancy triggers and applying ^^^, so we go up to 2^^^^(2^^^2^^^2^^^6). Then Ziatora, the Incinerator gets us up to 2^^^^2^^^^(2^^^2^^^2^^^6). The last Cadric trigger gets us that many Ziatoras, so the end step should get us to F_{w2}(2^^^^2^^^^(2^^^2^^^2^^^6)).
Edit: Whoops, that should be FF_{w2+1}(2^^^^2^^^^(2^^^2^^^2^^^6)).
Silverquill can give inklings to the opponent, and also each copy can give us a +1/+1 counter trigger on combat. This should let us get past Graham's Number before the end step, then feed the results back into our number of Ziatoras. We can then give the opponent more inklings later with Saw in Half. The downside is that, to my understanding, giving +1/+1 counters to all our creatures means that aside from Astral Dragon, anything we have out before combat will no longer be safe to target with Saw in Half. So I think we'll need to hold off on playing Toralf and Ziatora until after combat, and we won't be able to make more copies of Cadric, Soul Kindler after combat, but that should be fine since increased numbers of Primal Vigors will scale up our token generation regardless.
Thinking about it again, my concern wasn't the time between the draw triggers, it was that there's no way to get Saw back onto the library before the first draw trigger from a creature that can't help us topdeck. Which is fine because Temur Ascendancy is a may.
Play Avabruck Caretaker (6 life). Then play Mystic Sanctuary from hand to topdeck Saw and play it again (3 life). Turn the Caretaker into four tokens, getting a trigger from each Flayer (19 life).
Play Wingspan Mentor and Bloodthorn Taunter, followed by Astral Dragon (6 life). Copy the Sanctuary, topdeck the Saw, and play it on the dragon, going down to 3 life, then back up to 19.
Play Saw three more times on Astral Dragon, going to 10 life and 2^^^5 Doubling Seasons. At the end, get a couple of extra rounds of Mystic Sanctuaries to topdeck and play Saw two more times, on Avabruck Caretaker and Wingspan Mentor, ending up with 2^^^5 of each and 4 life, as well as Saw back on top of our deck.
Go to combat. Each Caretaker triggers, targeting a different Wingspan Mentor - except the first one, which will go to a Mystic Sanctuary token. After pumping the token, play Saw on it, splitting it and gaining another 16 life. At this point, continue like in a normal Saw in Half stage, gaining life every time we destroy a big token and using it to cast Saw more times on Astral Dragon and Sultai Flayer. Once the Sanctuary tokens get small, keep three of them at 5+ power. We'll have two more extra Caretaker triggers since we have three more Caretakers than Mentors, so we can then use those to repeat the process with Astral Dragons. Finish at 10+ life.
At this point, resolve the fourth trigger, pumping the first Wingspan Mentor. Cast Saw once on an Astral Dragon for more Doubling Seasons and topdeck triggers, using a topdeck trigger to cast Saw on Bloodthorn Taunter. Use four Taunters to give haste to the Mentor and three Sanctuaries, then activate the Mentor, pumping all of our fliers.
Each time we activate a Mentor, we can run a Saw stage with our fliers, splitting them until they get too small to give us life back, and spending any extra life on additional copying. (The Astral Dragons and the Sanctuary tokens have flying naturally, and we can also put flying tokens on the Flayers to pump them as well.) Then we can go back and split the Mentor more, getting more activations until its power gets too low to get haste from the Taunters, and playing out a Saw-Mentor stage in the process. Because we have no way to give flying to our Mentors, the only way for them to get big is resolving a Caretaker trigger, so the damage stays finite.
If this all works, I believe we end up at a damage of F{w2+1}(2^^^5).
Ooh, that's a really interesting idea. Getting two Saw in Half stages, divided into Human and non-Human versions. That really opens up possibilities!
The first Mystic Sanctuary comes into play tapped, so it its ability doesn't trigger, I think. So the opening needs to be revised.
Just looking at it initially, it looks like it should go infinite: We can spend 3 blue mana on Wingspan Mentor to give a lot of +1/+1 counters to our Mystic Sanctuarys. Then, Saw in Half on a big Mystic Sanctuary, getting a bunch of copies, and Saw in Half a Bloodthorn Taunter, allowing us to give a bunch of Mystic Sanctuaries haste for more blue mana. I guess we have to use Saw in Half on an Astral Dragon to be able to use Saw in Half on a Bloodthorn Taunter, but since Wingspan Mentor adds +1/+1 counters to our Astral Dragons as well, this should be no problem.
Edit: No, that doesn't quite work; the new Mystic Sanctuaries won't be creatures, even if they have specified base power and toughness. Still, we can cheaply create new Mystic Sanctuary creatures via Astral Dragon, and can turn them into large creatures using 3 blue mana via Wingspan Mentor, and then give them haste using Bloodthorn Taunter. So it still looks like an infinite.
Ooh, that's a really interesting idea. Getting two Saw in Half stages, divided into Human and non-Human versions. That really opens up possibilities!
The first Mystic Sanctuary comes into play tapped, so it its ability doesn't trigger, I think. So the opening needs to be revised.
Just looking at it initially, it looks like it should go infinite: We can spend 3 blue mana on Wingspan Mentor to give a lot of +1/+1 counters to our Mystic Sanctuarys. Then, Saw in Half on a big Mystic Sanctuary, getting a bunch of copies, and Saw in Half a Bloodthorn Taunter, allowing us to give a bunch of Mystic Sanctuaries haste for more blue mana. I guess we have to use Saw in Half on an Astral Dragon to be able to use Saw in Half on a Bloodthorn Taunter, but since Wingspan Mentor adds +1/+1 counters to our Astral Dragons as well, this should be no problem.
Edit: No, that doesn't quite work; the new Mystic Sanctuaries won't be creatures, even if they have specified base power and toughness. Still, we can cheaply create new Mystic Sanctuary creatures via Astral Dragon, and can turn them into large creatures using 3 blue mana via Wingspan Mentor, and then give them haste using Bloodthorn Taunter. So it still looks like an infinite.
Wingspan Mentor's ability doesn't just cost 3 blue mana, it also costs tapping a Wingspan Mentor with haste, which requires it having power 5+. So I don't see a way for that to go infinite?
As for copied Sanctuaries being creatures, it looks to me like them being creatures is a copiable attribute based on how Astral Dragon's ability is worded.
Play Saw on the Skaab to get a series of Exploit triggers. Between them, replay Saw 25 times, pointing the first 24 at Astral Dragon to get to 2^^^28, and the final one at Wingspan Mentor. Use the final Astral Dragon triggers to get 2^^^28 copies of Fight Rigging, and also 2^^^28 animated treasures. 9 (usable) treasures.
Go to combat, each Fight Rigging triggers to point at Wingspan Mentors. The original resolves first, letting us play Bloodthorn Taunter. Play Saw once on a Skaab to start an Exploit chain, then once on the Taunter, giving the Mentor haste so we can use the last three non-animated treasures to activate the Mentor and pump our animated ones. This lets us run the lower stage using the animated treasures, needing them to be at 5+ power to gain haste and tap for mana. Unfortunately we do have to sacrifice some of our big treasures rather than splitting them, but that shouldn't have a significant impact.
Well done! Saw in Half should make 64 Astral Dragons I think, increasing the number of Doubling Seasons to more than 2^^66. Then the next Saw in Half should make more than 2^^67 Repository Skaab, but I guess we're limited by mana. Hmm, but do we have to use a Saw on the Taunter before using it on the Mentor?
For the unlimited mana challenge, I think we can use the previous version, without Black Lotus and Channel, since we still need to pay life to Bolas's Citadel to get access to cards in our library.
Edit: I guess we would have to Saw the Taunter afterwards to get more mana, so it doesn't matter.
Play Fight Rigging, followed by Astral Dragon making four more Fight Rigging. The five of them hideaway the five remaining cards. 4 Treasures.
Go to combat. Each Fight Rigging triggers, targeting Astral Dragon. The first one just gets it to 6 power and fails to play Wingspan Mentor, while the next three play Finest Hour, Bloodthorn Taunter, and The Mirari Conjecture, which goes to two lore counters. The final one plays Saw in Half on the dragon, making four dragons with 7 power each. The first two go for Doubling Season, going to 1 > 5 > 69, while the third and fourth make 2^70 copies of Finest Hour and The Mirari Conjecture. The Conjecture copies max out on lore counters, getting back Saw and adding storm triggers.
Tap the Taunter to give haste to an Astral Dragon and attack, triggering all 2^70 copies of Finest Hour. Then play Saw, targeting the Taunter, and getting copied by each completed Conjecture. The copies go on the stack separately, first hitting the attacking dragon to keep it from killing the opponent, then splitting the copies to make more Doubling Seasons. The final three dragons should make clones of Fight Rigging, Conjecture, and the last Treasure token. Then the original Saw resolves, resupplying us with Taunters.
In the first of the 2^70 extra combats, use the Fight Riggings to run a Treasure stage, as well as finally playing the Wingspan Mentor. Then for future combats we can target the Mentors and run a proper double stage.
I'm not sure how to translate the extra combats to damage, but we should end up with whatever that gives us.
Very nice. Each trigger of Fight Rigging gets us a large Wingspan Mentor, so that's F_{w2}. Each combat triggers a lot of Fight Riggings, so that's F_{w2+1}. After the first extra combat, we get a lot of Doubling Seasons, but certainly less than F_{w2+1}(2^70). So we get F_{w2+1} from the remaining 2^70 - 1 combats, for F_{w2+2}(2^70 - 1).
Edit: Hmm, I wonder how many cards we would need to make a hyperstage?
Saw in Half makes for our most efficient stage, but it doesn't look like that would be good for a hyperstage. The resources for Saw in Half are our creatures on the battlefield, and their power/toughness. So we would have to use a sweeper to get rid of our creatures in the hyperstage transition, to avoid getting resources from the higher stage. But, that would leave us with no creatures in the lower stage, rather than reverting to what we had previously. So that seems to be no good.
Toralf, God of Fury is our next most efficient stage, and it looks more amenable to a hyperstage. Maybe very amenable, even - since it is so reliant on stack abilities, it's not obvious to me that we even need a sweeper in the hyperstage transition. That could potentially save some cards for recovery. We do need to make sure that we have to resolve the ability that deals more damage before the ability that gets our hyperstage resource back, or we can just repeat infinitely. Of course, a sweeper could do that, but it would be nice to avoid it. One thing that might work, is to cast the spell or activate the ability that gets our hyperstage resource back, and then the damage ability triggers off of the casting or activation. But, I can't think of anything that triggers off of a casting to deal damage.
Very nice. Each trigger of Fight Rigging gets us a large Wingspan Mentor, so that's F_{w2}. Each combat triggers a lot of Fight Riggings, so that's F_{w2+1}. After the first extra combat, we get a lot of Doubling Seasons, but certainly less than F_{w2+1}(2^70). So we get F_{w2+1} from the remaining 2^70 - 1 combats, for F_{w2+2}(2^70 - 1).
Edit: Hmm, I wonder how many cards we would need to make a hyperstage?
Saw in Half makes for our most efficient stage, but it doesn't look like that would be good for a hyperstage. The resources for Saw in Half are our creatures on the battlefield, and their power/toughness. So we would have to use a sweeper to get rid of our creatures in the hyperstage transition, to avoid getting resources from the higher stage. But, that would leave us with no creatures in the lower stage, rather than reverting to what we had previously. So that seems to be no good.
Toralf, God of Fury is our next most efficient stage, and it looks more amenable to a hyperstage. Maybe very amenable, even - since it is so reliant on stack abilities, it's not obvious to me that we even need a sweeper in the hyperstage transition. That could potentially save some cards for recovery. We do need to make sure that we have to resolve the ability that deals more damage before the ability that gets our hyperstage resource back, or we can just repeat infinitely. Of course, a sweeper could do that, but it would be nice to avoid it. One thing that might work, is to cast the spell or activate the ability that gets our hyperstage resource back, and then the damage ability triggers off of the casting or activation. But, I can't think of anything that triggers off of a casting to deal damage.
I'm still having a hard time understanding hyperstages. Would Sphinx-Bone Wand work?
If we need to keep something relevant to a Saw stage between iterations, could The Mirari Conjecture help?
Speaking of which, we should be able to improve the start of the previous deck by replacing Treasure Vault with Bucknard's Everfull Purse, netting an additional 5 mana so we can play Saw an additional time during combat. I think that gets us to 2^^^2^70 copies of Finest Hour?
A hyperstage is basically the same stage repeated X times on top of each other, rather than different stages. All the weird stuff we talk about, are just machinations to make sure it operates as it's supposed to, and doesn't go infinite. So we have a hyperstage resource, which functions just like the stage resource in resource-based stages; it limits how many times we can stack the stages on top of each other, and makes sure we can rebuild back up to X stages, no matter how far we resolve down. The stages are demarcated by "hyperstage transitions"; we have to make sure that we don't get back our hyperstage resource until we resolve down below the hyperstage transition, since if we get it back immediately, there's nothing stopping us from repaying it again and again. Also, we have to make sure that we can't bring the gains we make in the higher stage to the lower stage, so typically we have some "destructive" ability in the hyperstage that gets rid of anything that helps the stage, whether it be creature tokens, mana, or life. To make sure that we have to resolve the destructive ability, we tie that ability to getting our hyperstage resource back. Then, we also need a "beneficial" ability to build the stage on top, like Divine Congregation for a life stage, Rally the Righteous for a creature token stage, and so on. We have to make sure that we can only get the beneficial ability in the upper stage; the destructive ability getting rid of things can help out with this, e.g. there's no benefit to Divine Congregation after a [c}Worldfire[/c] is resolved.
The Toralf stage maybe not needing a sweeper puts a twist on this; we need a new reason why the beneficial ability has to go on top, but Sphinx-Bone Wand looks really good for this.
Another thing we need is, after we resolve a hyperstage transition, we need to be able to restart the transition while spending some amount of the lower stage, without having to spend to much. The expenditure can be as little as is possible; as long as we can only make a finite number of expenditures before the stage is exhausted, we are good. Resolving some fixed number of layers is not too much, e.g. if we have to resolve 9 layers, and lose an ability from the 10th, before we rebuild and restart the hyperstage, we are okay; but spending a stage resource is not okay, as that will flat out lose a layer (or a couple of layers), so an X-layer stage would be exhausted after X expenditures, which is obviously not a stage's worth of recursion.
For Toralf, we need to have to spend some of our Toralf triggers (or spend something that we get from Toralf triggers) in order to restart the hyperstage transition. But, we have to make sure we can't get that benefit from Toralf triggers in the higher stages. So, Aven Fisher looks like a decent possibility, maybe redrawing our instant/sorcery, but Byway Courier won't work, if we can keep Clue tokens from higher stages to use further down.
Hmmm... With an extra Saw in Half, I think we can use the first to target Doubling Season rather than Finest Hour, getting 2^70 + 69 Doubling Seasons (We still need to target The Mirari Conjecture to get it the Saw back.) Then, the next saw will create more than 2^2^70 Astral Dragons, which I think gets us up to more than 2^^^(2^2^70) copies of Finest Hour.
Hmm, I've run into a problem with the Toralf hyperstage plan. In order to cast the hyperstage spell at multiple hyperstage transitions, we need to counter it, so that it isn't stuck on the stack. We can still get the benefit, if we have one copy of Swarm Intelligence, say. But, then we can just not counter the spell, and wait for both copies to resolve, getting the benefit twice and winding up ahead one resource.
Play Black Lotus, Channel, Lich's Mirror. Spend all of our life, causing Lich's Mirror to shuffle everything back and put us back at 20. Replay Lotus and Mirror, get more mana, repeat until we have as much mana as we need. We'll be here a while.
After the final Mirror round, leave those three in the library and draw the other seven cards. Play Doubling Season, Astral Dragon making four Doubling Season, Codex Shredder, Citadel Siege on Khans, Wingspan Mentor, and Bloodthorn Taunter. Then play Saw in Half making 64 Astral Dragons, with the first 62 copying Doubling Season and the last ones copying Codex Shredder and Citadel Siege. Sacrifice the original Shredder, get back Saw, and play it on the Mentor.
Go to combat. All the Citadel Sieges trigger, with the first one targeting a Codex Shredder and the others each targeting a different Mentor. (Conveniently, we have one more Siege than Mentors.) Make a Shredder big, give it haste, sacrifice it for Saw, make more Taunters. From there, run the two stages as usual, with Wingspan Mentor and Codex Shredder.
Final damage of about F{w2+1}(2^^64), I think?
(If we want to cut down on the number of Lich's Mirror loops needed at the start, we can swap in Fight Rigging over Citadel Siege to give us access to Black Lotus copies later on. This will slightly lower our damage, but not by enough to affect the final numbers.)
Edit: I think we don't even need Fight Rigging for that, if we get the final Lich's Mirror use with one of our spells on the stack. We could also do it with an ability on the stack but I don't think we can carry over anything game-breaking that way.
Now that I think about it, old tech would probably work decently well here. If we add Dual Nature, along with a creature that adds +1/+1 on ETB or activation, then that should also get F_{w+2} with 10 cards, but probably a bigger input due to all the draws we get with Dual Nature. Then, a second creature that bounces the first on activation or ETB could get us to F_{w+4} with 11 cards. A third creature would get F_{w+6}, or if the first two creatures were activated abilities, we could add Rings of Brighthearth as well for F_{w+6}. Then three creatures with Rings of Brightheart could get F_{w+9}, and so on.
Edit: Oh, that old tech relied on Vedalken Orrery / Leyline of Anticipation of course. We could still get something with flickering, or with adding Leyline once we have enough cards in the deck.
2 Show and Tell
3 Omniscience
4 Temur Ascendancy
5 Doubling Season
6 Astral Dragon
7 Cadric, Soul Kindler
8 Mystic Sanctuary
9 Saw in Half
10 Toralf, God of Fury
11 Rite of Passage
12 Hunted Troll
13 Ziatora, the Incinerator
Black Lotus, Show and Tell, Omniscience, Temur Ascendancy, Doubling Season, Astral Dragon making four Ascendancies, Cadric, Soul Kindler triggering Ascendancy 5 times to draw most of our remaining cards.
Play Saw in Half on Astral Dragon, making 4 dragons and making the Doubling Season count go 1 > 5 > 37 > 2^37, then use the last to make 2^2^37 Ascendancies. Mystic Sanctuary puts back Saw. Hunted Troll gets 2^2^37 draw triggers, each time drawing Saw, playing it on Astral Dragon for more Doubling Seasons and one more round of Mystic Sanctuaries. At the end, make another round of Ascendancies and use the last Saw on Hunted Troll, making more 4-power tokens and repeating the process.
At the end of that, make another round of Ascendancies and use the last Saw on Cadric. Play Toralf, God of Fury, triggering the Ascendancies and the Cadrics, paying for the Cadric triggers with Mystic Sanctuary mana. Repeat the process for every Cadric trigger, finishing by making more Cadrics again. Then play Ziatora, the Incinerator and do the same thing.
Use the last draw to pick up Rite of Passage and play it. Go to end step, let the Ziatoras trigger, and throw something unimportant at one of the opponent's tokens. Run the Toralf combo, finishing by having the last point of damage hit one of our Astral Dragons to trigger Rite of Passage, putting counters on it. When that happens, run the Saw in Half loop on it, finishing by making more Rite of Passage for more counter triggers next time, as well as getting more copies of Toralf and Hunter Troll. At the end of a Toralf round, use the last counter trigger to make the next Ziatora hit even bigger. Finish by having the last Ziatora go for our opponent's face.
Final damage should be a function of the number of Ziatora triggers, although I'm not sure how to modify the f{w} part for this.
Couple of notes: Doubling Seasons should go 1 > 5 > 69 > 2^70, but we need to use one ETB of Astral Dragon to copy Mystic Sanctuary. We only really need to draw Hunted Troll with the Cadric draws, so we can draw and cast Saw in Half more times.
So four of the draws can draw Saw in Half, and one can draw Hunted Troll. The first draw and cast gets us 1 > 5 > 69 > 2^70 Doubling Seasons, then we copy Mystic Sanctuary to put Saw in Half on top of the library. The second Saw in Half creates more than 2^2^70 Astral Dragons, getting us to 2^^(2^2^70) >> 2^^^4 Doubling Seasons, and the next two Saw in Halfs get us to more than 2^^^6 Doubling Seasons and Temur Ascendancies.
We get more than 2^^^6 draws off of Hunted Troll, getting us to 2^^^2^^^6 Doubling Seasons, then the last Saw on Hunted Troll creates 2^^^2^^^6 power 4 Hunted Trolls, each triggering Ascendancies, getting us to 2^^^2^^^2^^^6.
We make a bunch of Cadrics and play Toralf, God of Fury. Each Cadric trigger gets us a bunch of Ascendancy triggers and applying ^^^, so we go up to 2^^^^(2^^^2^^^2^^^6). Then Ziatora, the Incinerator gets us up to 2^^^^2^^^^(2^^^2^^^2^^^6). The last Cadric trigger gets us that many Ziatoras, so the end step should get us to F_{w2}(2^^^^2^^^^(2^^^2^^^2^^^6)).
Edit: Whoops, that should be FF_{w2+1}(2^^^^2^^^^(2^^^2^^^2^^^6)).
Thinking about improvements from there, Doubling Season should be Primal Vigor but I don't think that's big enough to even show up in our numbers. A bigger one is that I think we can swap out Hunted Troll for Shadrix Silverquill.
Silverquill can give inklings to the opponent, and also each copy can give us a +1/+1 counter trigger on combat. This should let us get past Graham's Number before the end step, then feed the results back into our number of Ziatoras. We can then give the opponent more inklings later with Saw in Half. The downside is that, to my understanding, giving +1/+1 counters to all our creatures means that aside from Astral Dragon, anything we have out before combat will no longer be safe to target with Saw in Half. So I think we'll need to hold off on playing Toralf and Ziatora until after combat, and we won't be able to make more copies of Cadric, Soul Kindler after combat, but that should be fine since increased numbers of Primal Vigors will scale up our token generation regardless.
So that gets us to... let's see if I'm following the terms right. F{w2+1}(F{w+1}(2^^^^2^^^^2^^^6))?
2 Show and Tell
3 Bolas's Citadel
4 Doubling Season
5 Sultai Flayer
6 Saw in Half
7 Avabruck Caretaker
8 Mystic Sanctuary
9 Wingspan Mentor
10 Bloodthorn Taunter
11 Astral Dragon
Mulligan to 4. Play Black Lotus, Show and Tell, and Bolas's Citadel.
Flip Doubling Season and Sultai Flayer, going to 11 life. Then play Saw in Half on the Flayer, making four tokens and going down to 8, then up to 12.
Play Avabruck Caretaker (6 life). Then play Mystic Sanctuary from hand to topdeck Saw and play it again (3 life). Turn the Caretaker into four tokens, getting a trigger from each Flayer (19 life).
Play Wingspan Mentor and Bloodthorn Taunter, followed by Astral Dragon (6 life). Copy the Sanctuary, topdeck the Saw, and play it on the dragon, going down to 3 life, then back up to 19.
Play Saw three more times on Astral Dragon, going to 10 life and 2^^^5 Doubling Seasons. At the end, get a couple of extra rounds of Mystic Sanctuaries to topdeck and play Saw two more times, on Avabruck Caretaker and Wingspan Mentor, ending up with 2^^^5 of each and 4 life, as well as Saw back on top of our deck.
Go to combat. Each Caretaker triggers, targeting a different Wingspan Mentor - except the first one, which will go to a Mystic Sanctuary token. After pumping the token, play Saw on it, splitting it and gaining another 16 life. At this point, continue like in a normal Saw in Half stage, gaining life every time we destroy a big token and using it to cast Saw more times on Astral Dragon and Sultai Flayer. Once the Sanctuary tokens get small, keep three of them at 5+ power. We'll have two more extra Caretaker triggers since we have three more Caretakers than Mentors, so we can then use those to repeat the process with Astral Dragons. Finish at 10+ life.
At this point, resolve the fourth trigger, pumping the first Wingspan Mentor. Cast Saw once on an Astral Dragon for more Doubling Seasons and topdeck triggers, using a topdeck trigger to cast Saw on Bloodthorn Taunter. Use four Taunters to give haste to the Mentor and three Sanctuaries, then activate the Mentor, pumping all of our fliers.
Each time we activate a Mentor, we can run a Saw stage with our fliers, splitting them until they get too small to give us life back, and spending any extra life on additional copying. (The Astral Dragons and the Sanctuary tokens have flying naturally, and we can also put flying tokens on the Flayers to pump them as well.) Then we can go back and split the Mentor more, getting more activations until its power gets too low to get haste from the Taunters, and playing out a Saw-Mentor stage in the process. Because we have no way to give flying to our Mentors, the only way for them to get big is resolving a Caretaker trigger, so the damage stays finite.
If this all works, I believe we end up at a damage of F{w2+1}(2^^^5).
That part fails because you don't control three or more other Islands.
The first Mystic Sanctuary comes into play tapped, so it its ability doesn't trigger, I think. So the opening needs to be revised.
Just looking at it initially, it looks like it should go infinite: We can spend 3 blue mana on Wingspan Mentor to give a lot of +1/+1 counters to our Mystic Sanctuarys. Then, Saw in Half on a big Mystic Sanctuary, getting a bunch of copies, and Saw in Half a Bloodthorn Taunter, allowing us to give a bunch of Mystic Sanctuaries haste for more blue mana. I guess we have to use Saw in Half on an Astral Dragon to be able to use Saw in Half on a Bloodthorn Taunter, but since Wingspan Mentor adds +1/+1 counters to our Astral Dragons as well, this should be no problem.
Edit: No, that doesn't quite work; the new Mystic Sanctuaries won't be creatures, even if they have specified base power and toughness. Still, we can cheaply create new Mystic Sanctuary creatures via Astral Dragon, and can turn them into large creatures using 3 blue mana via Wingspan Mentor, and then give them haste using Bloodthorn Taunter. So it still looks like an infinite.
As for copied Sanctuaries being creatures, it looks to me like them being creatures is a copiable attribute based on how Astral Dragon's ability is worded.
2 Channel
3 Coveted Jewel
4 Doubling Season
5 Treasure Map
6 Astral Dragon
7 Saw in Half
8 Fight Rigging
9 Repository Skaab
10 Wingspan Mentor
11 Bloodthorn Taunter
Play Black Lotus and Channel. Go to 1 life, 19 C, 1 G.
Play Coveted Jewel, draw three cards, and tap it for UUU. Then play Doubling Season followed by Astral Dragon, making four more Doubling Seasons. 4 mana.
Play Treasure Map and activate it, putting 32 landmark counters on it. This transforms it into Treasure Cove and creates 96 treasure tokens. 1 mana.
Play Saw in Half, making 32 Astral Dragons and copying Doubling Season with each one. Then play Fight Rigging, Repository Skaab, and Wingspan Mentor, having the Skaab sacrifice a dragon token to get back Saw. Fight Rigging hits Bloodthorn Taunter with its hideaway. 84 treasures.
Play Saw on the Skaab to get a series of Exploit triggers. Between them, replay Saw 25 times, pointing the first 24 at Astral Dragon to get to 2^^^28, and the final one at Wingspan Mentor. Use the final Astral Dragon triggers to get 2^^^28 copies of Fight Rigging, and also 2^^^28 animated treasures. 9 (usable) treasures.
Go to combat, each Fight Rigging triggers to point at Wingspan Mentors. The original resolves first, letting us play Bloodthorn Taunter. Play Saw once on a Skaab to start an Exploit chain, then once on the Taunter, giving the Mentor haste so we can use the last three non-animated treasures to activate the Mentor and pump our animated ones. This lets us run the lower stage using the animated treasures, needing them to be at 5+ power to gain haste and tap for mana. Unfortunately we do have to sacrifice some of our big treasures rather than splitting them, but that shouldn't have a significant impact.
Overall damage F{w2+1}(2^^^28), I think?
For the unlimited mana challenge, I think we can use the previous version, without Black Lotus and Channel, since we still need to pay life to Bolas's Citadel to get access to cards in our library.
Edit: I guess we would have to Saw the Taunter afterwards to get more mana, so it doesn't matter.
2 Channel
3 Treasure Vault
4 Doubling Season
5 Finest Hour
6 Astral Dragon
7 Saw in Half
8 Fight Rigging
9 The Mirari Conjecture
10 Wingspan Mentor
11 Bloodthorn Taunter
Mulligan to 6. Play Black Lotus, Channel, Doubling Season, Treasure Vault, crack it with X=7 for 14 Treasures. 1 mana.
Play Fight Rigging, followed by Astral Dragon making four more Fight Rigging. The five of them hideaway the five remaining cards. 4 Treasures.
Go to combat. Each Fight Rigging triggers, targeting Astral Dragon. The first one just gets it to 6 power and fails to play Wingspan Mentor, while the next three play Finest Hour, Bloodthorn Taunter, and The Mirari Conjecture, which goes to two lore counters. The final one plays Saw in Half on the dragon, making four dragons with 7 power each. The first two go for Doubling Season, going to 1 > 5 > 69, while the third and fourth make 2^70 copies of Finest Hour and The Mirari Conjecture. The Conjecture copies max out on lore counters, getting back Saw and adding storm triggers.
Tap the Taunter to give haste to an Astral Dragon and attack, triggering all 2^70 copies of Finest Hour. Then play Saw, targeting the Taunter, and getting copied by each completed Conjecture. The copies go on the stack separately, first hitting the attacking dragon to keep it from killing the opponent, then splitting the copies to make more Doubling Seasons. The final three dragons should make clones of Fight Rigging, Conjecture, and the last Treasure token. Then the original Saw resolves, resupplying us with Taunters.
In the first of the 2^70 extra combats, use the Fight Riggings to run a Treasure stage, as well as finally playing the Wingspan Mentor. Then for future combats we can target the Mentors and run a proper double stage.
I'm not sure how to translate the extra combats to damage, but we should end up with whatever that gives us.
Edit: Hmm, I wonder how many cards we would need to make a hyperstage?
Saw in Half makes for our most efficient stage, but it doesn't look like that would be good for a hyperstage. The resources for Saw in Half are our creatures on the battlefield, and their power/toughness. So we would have to use a sweeper to get rid of our creatures in the hyperstage transition, to avoid getting resources from the higher stage. But, that would leave us with no creatures in the lower stage, rather than reverting to what we had previously. So that seems to be no good.
Toralf, God of Fury is our next most efficient stage, and it looks more amenable to a hyperstage. Maybe very amenable, even - since it is so reliant on stack abilities, it's not obvious to me that we even need a sweeper in the hyperstage transition. That could potentially save some cards for recovery. We do need to make sure that we have to resolve the ability that deals more damage before the ability that gets our hyperstage resource back, or we can just repeat infinitely. Of course, a sweeper could do that, but it would be nice to avoid it. One thing that might work, is to cast the spell or activate the ability that gets our hyperstage resource back, and then the damage ability triggers off of the casting or activation. But, I can't think of anything that triggers off of a casting to deal damage.
If we need to keep something relevant to a Saw stage between iterations, could The Mirari Conjecture help?
Speaking of which, we should be able to improve the start of the previous deck by replacing Treasure Vault with Bucknard's Everfull Purse, netting an additional 5 mana so we can play Saw an additional time during combat. I think that gets us to 2^^^2^70 copies of Finest Hour?
A hyperstage is basically the same stage repeated X times on top of each other, rather than different stages. All the weird stuff we talk about, are just machinations to make sure it operates as it's supposed to, and doesn't go infinite. So we have a hyperstage resource, which functions just like the stage resource in resource-based stages; it limits how many times we can stack the stages on top of each other, and makes sure we can rebuild back up to X stages, no matter how far we resolve down. The stages are demarcated by "hyperstage transitions"; we have to make sure that we don't get back our hyperstage resource until we resolve down below the hyperstage transition, since if we get it back immediately, there's nothing stopping us from repaying it again and again. Also, we have to make sure that we can't bring the gains we make in the higher stage to the lower stage, so typically we have some "destructive" ability in the hyperstage that gets rid of anything that helps the stage, whether it be creature tokens, mana, or life. To make sure that we have to resolve the destructive ability, we tie that ability to getting our hyperstage resource back. Then, we also need a "beneficial" ability to build the stage on top, like Divine Congregation for a life stage, Rally the Righteous for a creature token stage, and so on. We have to make sure that we can only get the beneficial ability in the upper stage; the destructive ability getting rid of things can help out with this, e.g. there's no benefit to Divine Congregation after a [c}Worldfire[/c] is resolved.
The Toralf stage maybe not needing a sweeper puts a twist on this; we need a new reason why the beneficial ability has to go on top, but Sphinx-Bone Wand looks really good for this.
Another thing we need is, after we resolve a hyperstage transition, we need to be able to restart the transition while spending some amount of the lower stage, without having to spend to much. The expenditure can be as little as is possible; as long as we can only make a finite number of expenditures before the stage is exhausted, we are good. Resolving some fixed number of layers is not too much, e.g. if we have to resolve 9 layers, and lose an ability from the 10th, before we rebuild and restart the hyperstage, we are okay; but spending a stage resource is not okay, as that will flat out lose a layer (or a couple of layers), so an X-layer stage would be exhausted after X expenditures, which is obviously not a stage's worth of recursion.
For Toralf, we need to have to spend some of our Toralf triggers (or spend something that we get from Toralf triggers) in order to restart the hyperstage transition. But, we have to make sure we can't get that benefit from Toralf triggers in the higher stages. So, Aven Fisher looks like a decent possibility, maybe redrawing our instant/sorcery, but Byway Courier won't work, if we can keep Clue tokens from higher stages to use further down.
Hmmm... With an extra Saw in Half, I think we can use the first to target Doubling Season rather than Finest Hour, getting 2^70 + 69 Doubling Seasons (We still need to target The Mirari Conjecture to get it the Saw back.) Then, the next saw will create more than 2^2^70 Astral Dragons, which I think gets us up to more than 2^^^(2^2^70) copies of Finest Hour.
Meanwhile, I believe I have a solution for two stages in 10 cards. Infinite mana is fine as long as the damage stays finite, right?
2 Channel
3 Lich's Mirror
4 Doubling Season
5 Astral Dragon
6 Codex Shredder
7 Citadel Siege
8 Saw in Half
9 Wingspan Mentor
10 Bloodthorn Taunter
Play Black Lotus, Channel, Lich's Mirror. Spend all of our life, causing Lich's Mirror to shuffle everything back and put us back at 20. Replay Lotus and Mirror, get more mana, repeat until we have as much mana as we need. We'll be here a while.
After the final Mirror round, leave those three in the library and draw the other seven cards. Play Doubling Season, Astral Dragon making four Doubling Season, Codex Shredder, Citadel Siege on Khans, Wingspan Mentor, and Bloodthorn Taunter. Then play Saw in Half making 64 Astral Dragons, with the first 62 copying Doubling Season and the last ones copying Codex Shredder and Citadel Siege. Sacrifice the original Shredder, get back Saw, and play it on the Mentor.
Go to combat. All the Citadel Sieges trigger, with the first one targeting a Codex Shredder and the others each targeting a different Mentor. (Conveniently, we have one more Siege than Mentors.) Make a Shredder big, give it haste, sacrifice it for Saw, make more Taunters. From there, run the two stages as usual, with Wingspan Mentor and Codex Shredder.
Final damage of about F{w2+1}(2^^64), I think?
(If we want to cut down on the number of Lich's Mirror loops needed at the start, we can swap in Fight Rigging over Citadel Siege to give us access to Black Lotus copies later on. This will slightly lower our damage, but not by enough to affect the final numbers.)
Edit: I think we don't even need Fight Rigging for that, if we get the final Lich's Mirror use with one of our spells on the stack. We could also do it with an ability on the stack but I don't think we can carry over anything game-breaking that way.