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I keep going back and forth on it, but I'm pretty sure sacellum doesn't work because of the higher toughness. You can attack with sacellum and minion. They have to block both, for a net loss of at least 2 spirits after you possibly ping gang blockers. Then you can untap both creatures by giving them one spirit. Eventually they'll be down to one token and have to let damage through.

EDIT: I think actually you can't do any of that because once you untap Pools, you can't untap the Orchard or the Raider anymore, so you can never attack safely with Minion. Someone else should check this over because I can't sit down with cards right now.

ED2: don't have time to elaborate now but I think i see a faster line by alternating attacks with the two creatures.

Switching to Juniper Order Druid and Icatian Store should solve the problems, but then we need a different creature-type match, and the only option I see is the earlier-mentioned Sutured Ghoul and Smokespew Invoker, which, although resulting in slower increases, should still get past 2^^42. (When working this out, keep in mind the possibility of finishing creatures off in response to Forbidden Orchard's triggered ability, thereby taking them down a little earlier; this possibility seems to have been missed in earlier analyses.)

X: Orchard, drain 2 counters from Store, Ley Druid

opponent X: attack for 1 (19)

X+1: drain 4 counters from Store, Intruder Alarm, tap Druid to untap Orchard, Smokespew Invoker

opponent X+1: attack for 3 (16)

X+2: drain 9 counters from Store, Coat of Arms with 4 floating

tap Orchard, give a token, Intruder Alarm trigger
response to trigger, tap Druid to untap Orchard
untap Druid and all opposing Spirits
repeat 2 more times (6 tokens in all)
Sutured Ghoul, get an Intruder Alarm trigger
untap Druid, then tap it again to untap Store (both lands are now untapped going into opponent's critical turn, as they must be)

opponent's X+2: attack for 36

drain 218 counters off Store, then kill things off with Invoker: 8, 11, 15, 21, 30, 44 tokens after each kill and the ensuing token trigger Druid must be untapped at the end of the turn

X+3: swing for 1 with Ghoul, get 44 blockers

drain 16,291,714,208 counters off Store, tap Druid to untap Store, and start getting colored mana
kill them all off again, deal 1 trample damage (19), opponent now has 2,327,387,744 tokens (this is between 2^^4 and 2^^5, and also between 1.5^^12 and 1.5^^13)

opponent X+3: over 1.5^^13 tokens

...

X+21: swing with Ghoul, kill all blockers, deal 1 trample damage (1), over 1.5^^48 tokens

opponent X+21: over 1.5^^49 tokens

X+22: swing with Ghoul and Invoker, ignore the blocker in front of Invoker, use over 1.5^^50 mana to kill all Ghoul blockers, deal 1 trample damage (0), win

1.5^^50 is less than 2^^42, but greater than 2^^41. Being more precise, though, thirty-eight 1.5s capped by a 2,327,387,744 is something that does exceed 2^^42, so the slower growth rate of Smokespew Invoker isn't a real concern at all even when starting from a number as low as 6.

and something else; I'm thinking of something that requires two colored mana from Forbidden Orchard and Spire of Industry, and that can either add life or make progress towards winning.

and one more card.
This one runs into trouble with the mana requirement of Tamiyo, the Moon Sage; Soulfire Grand Master and Tasigur, the Golden Fang can also serve the same purpose, but unfortunately they can win on their own.

This results in a large number of the opponent's tokens, but does not take a long time because the main process does not use any mana!
Note that the second dependency on Coat of Arms, from Sinstriker's Will, is necessary; we cannot rely on just the dependency from the last ability of Tezzeret, Agent of Bolas, because then it would be possible to accumulate loyalty counters without Coat of Arms, only playing it after getting enough.

and probably some planeswalker; each turn, the opponent's tokens will attack the planeswalker, and one has to block those that can be blocked, and sacrifice the blocking tokens for mana and for life from Plunge into Darkness, then use Murderous Betrayal to destroy the enemy tokens that were not blocked.

and something that requires red, green or white mana multiple times, and requires Progenitor Mimic to copy Doubling Season. Sapseep Forest would work if its ability were something useful; Goblin Kaboomist would work if the tokens could deal damage to players.

Mechanized Production could also be used if there is a way to make a single permanent an artifact that doesn't last just one turn.

Sheltered Valley
Saprazzan Cove
Forbidden Orchard
Sandstone Needle
Doubling Season
Argent Mutation
Ensoul Artifact
Progenitor Mimic
Sparkcaster

Endgame, we cast sparkcaster 20 turns in a row. Since Sandstone Needle and Forbidden Orchard are the only lands that let us cast it, we need to get at least 20 counters on Cove. So we need at least 4 doubling seasons (2^4=16) before we play it. The first time mimic (as doubling season) triggers, it generates 4 new doubling seasons, so we're good to go at that point (playing Cove, and casting sparkcaster the first time). Each cast of sparkcaster gives the opponent several spirits. I'll just run through the number of doubling seasons, for the damage dealt. We have 6 (1), then 2^6+6 (2), then 2^{2^6+6} (3), then 2 ^2^...(19 of these)...^2^6. But we don't worry about the ones we give out when dealing leathal damage, so we get attacked by 2^^18 spirits the turn before we win. We gain enough life to ride this out over the first several turns, so this takes 2^^18 turns. (Here, if it isn't clear, ^^ is two up arrows from Knuth's up-arrow notation)

Fractured identity is has possible use in being a smaller-number-of-cards way to produce lots of doubling seasons.

and one more card.
This one runs into trouble with the mana requirement of Tamiyo, the Moon Sage; Soulfire Grand Master and Tasigur, the Golden Fang can also serve the same purpose, but unfortunately they can win on their own.

We can add Faerie Conclave to this. It casts us tamiyo, and is a win condition through the army of spirits, that requires tapping the orchard to get through. We don't have enough mana to activate (that generic (1) in the cost is the only barrier).

Sheltered Valley
Saprazzan Cove
Forbidden Orchard
Sandstone Needle
Doubling Season
Argent Mutation
Ensoul Artifact
Progenitor Mimic
Sparkcaster

Endgame, we cast sparkcaster 20 turns in a row. Since Sandstone Needle and Forbidden Orchard are the only lands that let us cast it, we need to get at least 20 counters on Cove. So we need at least 4 doubling seasons (2^4=16) before we play it. The first time mimic (as doubling season) triggers, it generates 4 new doubling seasons, so we're good to go at that point (playing Cove, and casting sparkcaster the first time). Each cast of sparkcaster gives the opponent several spirits. I'll just run through the number of doubling seasons, for the damage dealt. We have 6 (1), then 2^6+6 (2), then 2^{2^6+6} (3), then 2 ^2^...(19 of these)...^2^6. But we don't worry about the ones we give out when dealing leathal damage, so we get attacked by 2^^18 spirits the turn before we win. We gain enough life to ride this out over the first several turns, so this takes 2^^18 turns. (Here, if it isn't clear, ^^ is two up arrows from Knuth's up-arrow notation)

That doesn't work, for several reasons:

Progenitor Mimic can copy the Spirit tokens.

Doubling Season applies only to tokens you control. (Primal Vigor doesn't, but it comes with another restriction that makes it not work here.)

and two more cards for mana production.
The sequence with that is Quest for Ancient Secrets, Parallel Lives, Fractured Identity on Parallel Lives (1 counter on Quest for Ancient Secrets), Pull from Eternity on Parallel Lives (3 counters), Vengeful Rebirth on Pull from Eternity, Pull from Eternity on Vengeful Rebirth (5 counters), activate Quest for Ancient Secrets.
But I haven't found any choice of mana producers that makes this take many turns, and even if there is one, it still just gives a tetration of less than 20.

Turn X: Mercurial Transformation on Doubling Season, Progenitor Mimic copying it, using mana from Sapseep Forest and Calciform Pools
Turn X+1: In upkeep, in response to Progenitor Mimic's and Timberline Ridge's triggered abilities, tap Timberline Ridge for G, adding 4 depletion counters, to activate Sapseep Forest, going to 21 life.
Repeat once the depletion counters run out. If the number of depletion counters is N once, it will exceed 2^^N the next time. 4 = 2^^^2, then 29 more iterations, so the final delay exceeds 2^^^31.

It looks like this assumes that the 7 cards are already on the battlefield, but the original rules say that they start in your hand. So for example, you would never play Grip of Chaos, if it only slowed you down.

Starting on the battlefield is perhaps the more interesting challenge, what with mana costs and all.

Wow, forced creation of Doubling Season tokens with a depletion land is a brilliant combo!

Unfortunately the hexation with Grip of Chaos won't work. That hand wouldn't win at all because Keldon Megaliths never hits the opponent.

By my count the Sapseep Forest version "only" needs 2^^^30 turns. We don't need to wait for the depletion counters after the last activation. Test of Endurance just wins after reaching 50 life. Still amazingly high numbers

Edit: Slithering Shade almost works as wincon for the hexation hand, but we'd need BG depletion and UR storage lands to cast everything. They are not available in those combinations.

I think it is more than 2^^^40 for the hand with 2x Aria of Flame. After casting all of our enchantments and activating Keldon Megaliths, we will put two depletion counters on Timberline Ridge, since we have one Doubling Season in play. So we have to wait three turns to use Timberline Ridge again, by which time we will have 2059 Doubling Seasons (obvious copying Aria of Flame is bad), and add 2^2059 > 2^^^3 depletion counters to Timberline Ridge. 37 damage pings later, we will add 2^^^40 depletion counters, and will have to wait more than 2^^^40 turns for the final damage.

I was looking at other ways to get the mana we need with Slithering Shade, unfortunately whichever of U or R we get with the depletion land, we'll need two of the other, along with two other colors, so that doesn't quite work out. I thought maybe an artifact could be copied by Extravagant Replication, but we need 6 mana for ER, that seems tough to make work.

Edit: Just for fun, suppose we looked at the game probabilistically rather than worst-case with Grip of Chaos. The Keldon Megaliths pings are guaranteed to kill after at most 39 pings. In terms of game length, the number for the worst-case scenario is much, much more than the odds against it happening:

Each turn, the worst-case scenario is to copy Doubling Season for all of the Extravagant Replication triggers except the last, where we target Extravagant Replication. (except for the turn we tap Timberline Ridge, where we just target Doubling Season.) If there are N Doubling Seasons before the last ER trigger, then there is a more than 1/N chance that we will target ER for the last creation of tokens. There are also all the random events from before, but they all factor in much less, so the total odds so far are 1/(CN) with C << N. Then, the next turn we want to first target Doubling Season with about 2^N ER's, which is about N/(2^N), for 1/(C*2^N) combined. Then the next target of Doubling Season will be about 1/2, then the rest will be overwhelmingly likely. So a 1/(2C*2^N) likelyhood, and we will get about 2^^(2^N) counters on Timberline Ridge after the second to last ping. So the expected number of turns required is about 2^^(2^N)/(2C*2^N) with C << N, i.e. multiplying by the probability is practically infinitesimal. After the first ping Timberline Ridge will get more than 2^^(2^^6) > 2^^^^3 depletion counters, so after the second to last it will get over 2^^^^40, so the expected number of turns will be more than 2^^^^40. Of course, the probability that we win is 50%.

We can activate Keldon Megaliths in response to the Extravagant Replication triggers. So the second ping would only give us 2^11 depletion counters, which is a bit short of the 2^16 we need to claim 2^^^40 in the end.

The idea was that using Lotus Blossom requires all the remaining cards to be cast in one turn. Casting Cruel Entertainment triggers Aria of Flame, which deals 2 damage because of Doubling Season, but then the opponent gets to take control for one turn, and can make Extravagant Replication target Aria of Flame, producing two copies because of Doubling Season.

The problem is that Lotus Blossom produces only one color of mana. The lands can make RG, but that still leaves 16UUB needed.

The idea here is that the only way to get a win condition into play is Rise of the Dark Realms, and the only way to get creatures into the graveyard for that is Monomania, which puts all our creatures into the graveyard since we need to keep Rise. So to win we have to donate the Resplendent Mentor + Famished Paladin combo via Sky Swallower.

We can break that combo up with Phthisis. But we can't keep it in hand because of Monomania so we need to have it suspended before Monomania. And we can't cast Rise in response to the suspend trigger so the Phthisis cast will only happen in our upkeep a turn later. That means the opponent had 1 turn to gain an arbitrary amount of life. After taking out Resplendent Mentor the opponent is left with a Famished Paladin that can attack once but never kill us. Meanwhile we have a Sky Swallower that attacks for 8 each turn and will eventually take them down.

So we have a winning strategy but the opponent can choose the number of turns it takes us to win. No finite bound on the number of turns is big enough to guarantee our win has happened in that time! We have to resort to ordinals and say we have a win in omega!

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EDIT: I think actually you can't do any of that because once you untap Pools, you can't untap the Orchard or the Raider anymore, so you can never attack safely with Minion. Someone else should check this over because I can't sit down with cards right now.

ED2: don't have time to elaborate now but I think i see a faster line by alternating attacks with the two creatures.

response to trigger, tap Druid to untap Orchard

untap Druid and all opposing Spirits

repeat 2 more times (6 tokens in all)

Sutured Ghoul, get an Intruder Alarm trigger

untap Druid, then tap it again to untap Store (

both lands are now untapped going into opponent's critical turn, as they must be)Druid must be untapped at the end of the turnkill them all off again, deal 1 trample damage (19), opponent now has 2,327,387,744 tokens (this is between 2^^4 and 2^^5, and also between 1.5^^12 and 1.5^^13)

1.5^^50 is less than 2^^42, but greater than 2^^41. Being more precise, though, thirty-eight 1.5s capped by a 2,327,387,744 is something that does exceed 2^^42, so the slower growth rate of Smokespew Invoker isn't a real concern at all even when starting from a number as low as 6.

Still trying to exponentiate the opponent's creatures every turn:

A small change to a previous deck:

- Forbidden Orchard

- Spire of Industry

- Rushwood Grove

- Coat of Arms

- Tamiyo, the Moon Sage

- Mercy Killing

and something else; I'm thinking of something that requires two colored mana from Forbidden Orchard and Spire of Industry, and that can either add life or make progress towards winning.- Forbidden Orchard

- Rushwood Grove

- Tamiyo, the Moon Sage

- Parallel Evolution

- Quiet Contemplation

- Leyline of Anticipation

and one more card.This one runs into trouble with the mana requirement of Tamiyo, the Moon Sage; Soulfire Grand Master and Tasigur, the Golden Fang can also serve the same purpose, but unfortunately they can win on their own.

- Forbidden Orchard

- Calciform Pools

- Tidewater Minion

- Intruder Alarm

- Coat of Arms

- Sinstriker's Will

- Tezzeret, Agent of Bolas

This results in a large number of the opponent's tokens, but does not take a long time because the main process does not use any mana!Note that the second dependency on Coat of Arms, from Sinstriker's Will, is necessary; we cannot rely on just the dependency from the last ability of Tezzeret, Agent of Bolas, because then it would be possible to accumulate loyalty counters without Coat of Arms, only playing it after getting enough.

- Fountain of Cho

- Alliance of Arms

- Phyrexian Altar

- Murderous Betrayal

- Plunge into Darkness

- Seasons Past

and probably some planeswalker; each turn, the opponent's tokens will attack the planeswalker, and one has to block those that can be blocked, and sacrifice the blocking tokens for mana and for life from Plunge into Darkness, then use Murderous Betrayal to destroy the enemy tokens that were not blocked.Taking a different direction:

- Saprazzan Cove

- Doubling Season

- Argent Mutation

- March of the Machines

- Progenitor Mimic

and something that requires red, green or white mana multiple times, and requires Progenitor Mimic to copy Doubling Season. Sapseep Forest would work if its ability were something useful; Goblin Kaboomist would work if the tokens could deal damage to players.Mechanized Production could also be used if there is a way to make a single permanent an artifact that doesn't last just one turn.

Sheltered Valley

Saprazzan Cove

Forbidden Orchard

Sandstone Needle

Doubling Season

Argent Mutation

Ensoul Artifact

Progenitor Mimic

Sparkcaster

Endgame, we cast sparkcaster 20 turns in a row. Since Sandstone Needle and Forbidden Orchard are the only lands that let us cast it, we need to get at least 20 counters on Cove. So we need at least 4 doubling seasons (2^4=16) before we play it. The first time mimic (as doubling season) triggers, it generates 4 new doubling seasons, so we're good to go at that point (playing Cove, and casting sparkcaster the first time). Each cast of sparkcaster gives the opponent several spirits. I'll just run through the number of doubling seasons, for the damage dealt. We have 6 (1), then 2^6+6 (2), then 2^{2^6+6} (3), then 2 ^2^...(19 of these)...^2^6. But we don't worry about the ones we give out when dealing leathal damage, so we get attacked by 2^^18 spirits the turn before we win. We gain enough life to ride this out over the first several turns, so this takes 2^^18 turns. (Here, if it isn't clear, ^^ is two up arrows from Knuth's up-arrow notation)

Fractured identity is has possible use in being a smaller-number-of-cards way to produce lots of doubling seasons.

We can add Faerie Conclave to this. It casts us tamiyo, and is a win condition through the army of spirits, that requires tapping the orchard to get through. We don't have enough mana to activate (that generic (1) in the cost is the only barrier).

That doesn't work, for several reasons:

I did have a look at that too, but the exiling makes it difficult. The best I could come up with is

- Parallel Lives

- Quest for Ancient Secrets

- Fractured Identity

- Pull from Eternity

- Vengeful Rebirth

and two more cards for mana production.The sequence with that is Quest for Ancient Secrets, Parallel Lives, Fractured Identity on Parallel Lives (1 counter on Quest for Ancient Secrets), Pull from Eternity on Parallel Lives (3 counters), Vengeful Rebirth on Pull from Eternity, Pull from Eternity on Vengeful Rebirth (5 counters), activate Quest for Ancient Secrets.

But I haven't found any choice of mana producers that makes this take many turns, and even if there is one, it still just gives a tetration of less than 20.

Turn X: Mercurial Transformation on Doubling Season, Progenitor Mimic copying it, using mana from Sapseep Forest and Calciform Pools

Turn X+1: In upkeep, in response to Progenitor Mimic's and Timberline Ridge's triggered abilities, tap Timberline Ridge for G, adding 4 depletion counters, to activate Sapseep Forest, going to 21 life.

Repeat once the depletion counters run out. If the number of depletion counters is N once, it will exceed 2^^N the next time. 4 = 2^^^2, then 29 more iterations, so the final delay exceeds 2^^^31.

Extravagant Replication + Doubling Season + Keldon Megaliths + Timberline Ridge + Saprazzan Cove gives >2^^^21 turns in only 5 cards. Add a second Extravagant Replication and a Grip of Chaos (relying on randomness giving the slowest possibility -- is this correct?) to get hexation.

Starting on the battlefield is perhaps the more interesting challenge, what with mana costs and all.

No, it doesn't. As in some previous versions, it is necessary to play all cards in order to enable Keldon Megaliths.

Unfortunately the hexation with Grip of Chaos won't work. That hand wouldn't win at all because Keldon Megaliths never hits the opponent.

By my count the Sapseep Forest version "only" needs 2^^^30 turns. We don't need to wait for the depletion counters after the last activation. Test of Endurance just wins after reaching 50 life. Still amazingly high numbers

Edit:

Slithering Shade almost works as wincon for the hexation hand, but we'd need BG depletion and UR storage lands to cast everything. They are not available in those combinations.

Adding 2x Aria of Flame to plopfill's 5 cards (Extravagant Replication, Doubling Season, Keldon Megaliths, Timberline Ridge, Saprazzan Cove) would get us to 2^^^39.

I was looking at other ways to get the mana we need with Slithering Shade, unfortunately whichever of U or R we get with the depletion land, we'll need two of the other, along with two other colors, so that doesn't quite work out. I thought maybe an artifact could be copied by Extravagant Replication, but we need 6 mana for ER, that seems tough to make work.

Edit: Just for fun, suppose we looked at the game probabilistically rather than worst-case with Grip of Chaos. The Keldon Megaliths pings are guaranteed to kill after at most 39 pings. In terms of game length, the number for the worst-case scenario is much, much more than the odds against it happening:

Each turn, the worst-case scenario is to copy Doubling Season for all of the Extravagant Replication triggers except the last, where we target Extravagant Replication. (except for the turn we tap Timberline Ridge, where we just target Doubling Season.) If there are N Doubling Seasons before the last ER trigger, then there is a more than 1/N chance that we will target ER for the last creation of tokens. There are also all the random events from before, but they all factor in much less, so the total odds so far are 1/(CN) with C << N. Then, the next turn we want to first target Doubling Season with about 2^N ER's, which is about N/(2^N), for 1/(C*2^N) combined. Then the next target of Doubling Season will be about 1/2, then the rest will be overwhelmingly likely. So a 1/(2C*2^N) likelyhood, and we will get about 2^^(2^N) counters on Timberline Ridge after the second to last ping. So the expected number of turns required is about 2^^(2^N)/(2C*2^N) with C << N, i.e. multiplying by the probability is practically infinitesimal. After the first ping Timberline Ridge will get more than 2^^(2^^6) > 2^^^^3 depletion counters, so after the second to last it will get over 2^^^^40, so the expected number of turns will be more than 2^^^^40. Of course, the probability that we win is 50%.

Lotus Blossom, Extravagant Replication, Doubling Season, Aria of Flame,Cruel Entertainment.The idea was that using Lotus Blossom requires all the remaining cards to be cast in one turn. Casting Cruel Entertainment triggers Aria of Flame, which deals 2 damage because of Doubling Season, but then the opponent gets to take control for one turn, and can make Extravagant Replication target Aria of Flame, producing two copies because of Doubling Season.

The problem is that Lotus Blossom produces only one color of mana. The lands can make RG, but that still leaves 16UUB needed.

- Bottomless Vault

- Monomania

- Rise of the Dark Realms

- Sky Swallower

- Resplendent Mentor

- Famished Paladin

- Phthisis

The idea here is that the only way to get a win condition into play is Rise of the Dark Realms, and the only way to get creatures into the graveyard for that is Monomania, which puts all our creatures into the graveyard since we need to keep Rise. So to win we have to donate the Resplendent Mentor + Famished Paladin combo via Sky Swallower.We can break that combo up with Phthisis. But we can't keep it in hand because of Monomania so we need to have it suspended before Monomania. And we can't cast Rise in response to the suspend trigger so the Phthisis cast will only happen in our upkeep a turn later. That means the opponent had 1 turn to gain an arbitrary amount of life. After taking out Resplendent Mentor the opponent is left with a Famished Paladin that can attack once but never kill us. Meanwhile we have a Sky Swallower that attacks for 8 each turn and will eventually take them down.

So we have a winning strategy but the opponent can choose the number of turns it takes us to win. No finite bound on the number of turns is big enough to guarantee our win has happened in that time! We have to resort to ordinals and say we have a win in omega!