The exponent for the Primal Vigor version ends up being 3, not 4. The turn you reduce the opponent to 17 life, I have the minimal opponent's board at 383 elves and 3 spirits, so that deck figures to get to about 3^^18 (you only have to play Mercy Killing once on the last turn).
Transcendence gets to 2^^18, thanks mostly to the acrobatics that are required to get the damn enchantment safely onto the board without dying. By taking the opponent down to 15 life, they get boosted up to 838,860,802 elves and one spirit. Naturally, this is faster than a 3^^18 turn clock.
The exponent for the Primal Vigor version ends up being 3, not 4. The turn you reduce the opponent to 17 life, I have the minimal opponent's board at 383 elves and 3 spirits, so that deck figures to get to about 3^^18 (you only have to play Mercy Killing once on the last turn).
You are definitely right that the base is 3.
My count on the tokens is a little different though. This is what I have for the start:
Your turn: Tap Orchard, kill spirit, cast Primal Vigor, (1 elf), tap down elf
Their turn: nothing
Your turn: Ping opponent to 19 (2 spirits, 1 elf), tap down spirit.
Their turn: attack with 1 elf, 1 spirit (17 life)
Your turn: Ping opponent to 18 (4 spirits, 1 elf), tap down a spirit.
Their turn: attack with 1 elf, 3 spirits (4 life)
Your turn: Ping opponent to 17 (6 spirits, 1 elf), tap down an elf
Their turn: attack, kill elf then 5 spirits (1 spirit gets through) (1 spirits, 41 elves, 3 life)
Your turn: Ping opponent to 16 (3 spirits, 41 elves), tap down an elf.
Their turn: attack, kill 40 elves then 2 spirits, (1 spirit, ~40*3^40 elves, 2 life)
So that's about 40*3^40 (> 3^3^3 i.e. 3^^3) casts to deal 5 damage, so I end up agreeing on 3^^18 anyway.
Transcendence gets to 2^^18, thanks mostly to the acrobatics that are required to get the damn enchantment safely onto the board without dying. By taking the opponent down to 15 life, they get boosted up to 838,860,802 elves and one spirit. Naturally, this is faster than a 3^^18 turn clock.
Here's what I have so far with Transcendence:
Your turn: Transcendence, response ping self (1 spirit), tap down spirit
Their turn: Kill the spirit (1 elf token)
Your turn: Ping opponent to 19 (1 spirit, 1 elf), tap down spirit.
Their turn: attack with elf, kill the elf (2 spirit, 1 elf)
Your turn: Ping opponent to 18 (2 spirit, 1 elf), tap down a spirit.
Their turn: attack with elf, spirit, kill both (1 spirit, 3 elves)
Your turn: Ping opponent to 17 (2 spirits, 3 elves), tap down an elf
Their turn: attack with 2 spirits, 2 elves, kill all (12 elves)
Your turn: Ping opponent to 16 (1 spirit, 12 elves), tap down an elf
Their turn: attack with 1 spirit, 11 elves, kill all (22,530 elves)
22,530 is less than 2^^4, so the lower bound is 2^^3 casts for 5 damage, and again I agree that's somewhat higher than 2^^18.
Static Orb might be better than Primal Vigor since it means I can only ping every other turn, but again I haven't done the math on it yet.
The entire hand has to be empty to use Megaliths, so on the Transcendence turn you have to do this:
-float R with Orchard, give them a spirit (2 cards in hand)
-play Transcendence (1 card in hand)
-respond to the spell with Mercy Killing, targeting the spirit (0 cards in hand)
-respond to both by using the floating mana for Megaliths, targeting self to 19
-Mercy Killing turns spirit into elf
-Tamiyo emblem returns Mercy Killing to hand
-Transcendence resolves
-Tap down the elf (not a spirit)
Except for that one-off deal (and the final turn where you shoot them to 0 right after untapping), you absolutely cannot afford to activate Megaliths on your own turn, since that will necessarily leave all your lands tapped, with no way of drawing more mana on their turn to play Mercy Killing and survive the attack. The following turns look like this:
-on their end step, Mercy Kill their elf, giving back 1 elf, and respond to the Tamiyo trigger with Megaliths to 19, giving them 1 spirit
-your turn, just lock down the spirit
-on their attack, kill their elf and replace it with another elf, then respond with Megaliths to 18 and 2 spirits
-again, lock down a spirit
-their attack, kill elf (1) followed by spirit (2), respond to the last return trigger with Megaliths to 17: 3 elves, 2 spirits
I was trying to find a way to do without Tamiyo, such as Reiterate + Remand, but those color requirements are crazy and leave nothing that could fit as a win condition.
The fact that Megaliths alone lends itself to a 6-card deck seemed promising for Cyclone along with something like Glimmerpost, but that forces no obligation to drop Coat of Arms (so Cyclone's payments will in fact kill all the tokens that get created). Megaliths are just too versatile to be cleanly replaced by either another land, or another win condition.
Static Orb will get to 2^^29, though, jumping far out into the lead.
Jeeze, Static Orb makes this even more ridiculous. Great job!
If you replace Tamiyo with Reiterate and Calciform Pools with Fungal Reaches does that work out? Unfortunately, you have to find some way to force the Forbidden Orchard to tap, which probably makes it worse in the end since you can't use Primal Vigor or Static Orb.
Static Orb will get to 2^^36, though, jumping far out into the lead.
If you've already worked it out, would you mind posting your work? I think we're looking at 900ish tokens on the turn we ping the opponent to 14, which would mean a turncount between 2^^31 and 2^^32, but I could have messed that up.
Jeeze, Static Orb makes this even more ridiculous. Great job!
If you replace Tamiyo with Reiterate and Calciform Pools with Fungal Reaches does that work out? Unfortunately, you have to find some way to force the Forbidden Orchard to tap, which probably makes it worse in the end since you can't use Primal Vigor or Static Orb.
The trouble with Reiterate by itself is eventually the turn will have to pass, and the original Mercy Killing resolves. When it does, there's no way to bring it back to hand, so you're out of luck for surviving the next strike. We need Remand to get everything back in hand (Reiterate the Remand twice, targeting the non-buyback spells), so that's a commitment to three different colors of mana every turn. Storage lands don't do that.
If you've already worked it out, would you mind posting your work? I think we're looking at 900ish tokens on the turn we ping the opponent to 14, which would mean a turncount between 2^^31 and 2^^32, but I could have messed that up.
That looks right--for some reason I had the token count after the first ping listed as 1:1, instead of 1:0, which threw off everything as far as safe attacks. The token count still looks to be a lot more than 900 going against 14, so here's what I have now:
-lands, charge (4 in hand)
-Tamiyo, Orb, Coat (1 in hand)
-untap, pass turn since there's nothing else productive to do
-opponent's next end step, tap Orchard for R (1 spirit), tap Pools to draw 4 of its counters into mana, Mercy Killing on the spirit (0 in hand), and respond with Megaliths to 19. The spirit gets replaced by an elf.
-untap two lands, lock down the elf
-untap the last land, lock it down again
-end step, Mercy Killing to spin the elf into another elf, and respond with Megaliths to 18 (1 elf, 1 spirit)
-untap 2, lock down the spirit this time
-take 1 from elf, to 19
-untap last land, lock down spirit again
-Mercy Killing to spin the attacking elf into another elf, Megaliths to 17 (1 elf, 2 spirits)
-untap 2, lock a spirit
-take 3 damage to 16
-untap last, lock spirit
-Mercy Killing only once on the attacking spirit, Megaliths to 16, take 2 from attacking elf to 14 (2 elves, 2 spirits)
-untap 2, lock something
-take 6 damage to 8
-untap last, lock elf
-Mercy Killing once on a spirit, Megaliths to 15, take 6 from remaining attackers to 2 (4 elves, 2 spirits)
-untap Megaliths and Orchard, lock spirit
-tap Megaliths to draw from Pools, Mercy Killing five times to survive the attack: 4, 7, 13, 25, 2 (27 elves, 2 spirits)
-untap Megaliths and Pools, lock spirit
-Mercy Killing on all 28 attackers: 27, 53, 105, ..., 1744830465, 2, Megaliths to 14 (3489660929 elves, 2 spirits, > 2^^4)
-2^^5 elves
-2^^6 elves, Megaliths to 13
...
-2^^30 elves, Megaliths to 1
-untap 2 lands
-draw from pools, 2^^31
-untap all lands, Mercy Killing once, respond with Megaliths to 0
It would be great if there was another card that functioned from the graveyard, since that would allow Static Orb to give way to Mist of Stagnation, with the motivation that doing it that way would require Tamiyo to ultimate from exactly 8 counters to become the required second card in the graveyard, and leave her unable to lock something down every turn.
I think that if we don't use Mercy Killing on any elves for as long as possible, we actually kill significantly faster. I made a spreadsheet to keep track of what it going on.
On the ping-15 turn, it's better to take the damage from all three and kill something after combat; that way they only get in for 7 instead of 8. This allows the waiting turn between 18 and 17 to become "they attack for 1, just take it", and that's one fewer copy of Mercy Killing that needs to be charged up for.
Numbers in the E273 range are certainly larger than 2^^3, but they're also larger than 2^^4, which is a relatively tame 65536. It's inconvenient to start counting from 2^^5 since the numbers have to be at least E19728, so that's a long way off, and here the final figure works out over 2^^29.
Steam Vines could work well as the extra card. It requires one to take no damage from anything else. Also, it cannot be attached to Calciform Pools (because that results in the loss of the storage counters) until after the penultimate shot from Keldon Megaliths, which means that it has to alternate between Keldon Megaliths and Forbidden Orchard most of the time, which introduces a delay once every three turns because Keldon Megaliths enters the battlefield tapped. I have not worked this out properly yet, but would guess that it comes close to the result with Static Orb.
Steam Vines appears to make the game unwinnable. It demands a sorcery-speed RR which must come from Megaliths and Orchard, which means that mana cannot be spent to pull counters off Pools (and because of the Orchard tap, the opponent will have at least one token). The third mana in the cost will require tapping Calciform Pools, so there can be no access to mana during the opponent's next turn, and they will be able to attack for at least 1, ensuring a lead in the damage race that Megaliths are too slow to ever overcome.
If it cost 2R instead, then it would be simple to drop it and Tamiyo without playing Orchard at all, let both lands get destroyed and returned, and decline the Tamiyo trigger to return Steam Vines. It sits harmlessly in the graveyard for the rest of the game while the opponent has no creatures, and then the real charging phase can begin.
Is final number 2^^29 or 2^^36? (Knuth's up-arrow notation?)
Also this is nuts.
The final number is somewhat greater that 2^^29. The spreadsheet I was linking to went away, so I updated the link in case anyone wants to see the math. A tighter lower bound would be f(27), where f(0)=7 and f(n+1)=n*2^n.
Steam Vines appears to make the game unwinnable. It demands a sorcery-speed RR which must come from Megaliths and Orchard, which means that mana cannot be spent to pull counters off Pools (and because of the Orchard tap, the opponent will have at least one token). The third mana in the cost will require tapping Calciform Pools, so there can be no access to mana during the opponent's next turn, and they will be able to attack for at least 1, ensuring a lead in the damage race that Megaliths are too slow to ever overcome.
Can't one simply use Tamiyo, the Moon Sage to stop the token from attacking?
Another idea I had is Quicksilver Fountain, which creates 3-turn periods during which only two can be used to activate Keldon Megaliths' ability; only one of them can be used once life is low enough that Mercy Killing is required every turn. However, if the opponent has enough cards replaced by basic lands, the opponent can play four lands, delaying the unflooding and making white mana unavailable once every four turns, and then there is no way to win.
Interestingly, the test deck can use Adventurers' guildhouse to give Derevi banding with Krenko. Luckily, it doesn't matter.
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Some facts of magic:
-Terror is an emotion which, when experienced, results in death.
-The pox was a disease notorious for having killed one-third, rounded up, of Europe’s population. Smallpox, on the other hand, killed only a single person.
-A person riding a horse cannot be stopped by foot soldiers, large animals, walls, archers, or even catapults.
Huh, in that case the game goes like this:
t1 Megaliths
t2 Pools
t7 Tamiyo
t12 emblem
t13 Orchard, Steam Vines (on Megaliths), give them a single token and lock it down with Tamiyo
t14 tap Megaliths, move Steam Vines onto Pools, tap Pools, drain it of its 5 counters, move Steam Vines onto Orchard, tap Orchard with Tamiyo (they do not get another token from this), use Tamiyo's emblem to get back all three lands but let Steam Vines stay in the graveyard for the rest of the game, replay Megaliths, and finally use the floating mana to play Coat of Arms while you still have the chance
opponent's t14 attack for 1, just take it (go to 16)
t15 Pools, start charging, start locking down the Spirit token again
etc.
and, with 16 life still in reserve at this point, it's just 2^^17.
Forbidden Orchard is used four times to charge Calciform Pools in the opponent's turns and once more to cast Tidewater Minion, which then stops the opponent from winning with the five spirits, and is used to charge Calciform Pools for a long time. Eventually, Ghost-Lit Raider is cast, and on the next turn, Intruder Alarm, Coat of Arms, and Splinterfright are cast. From then on, Ghost-Lit Raider, Forbidden Orchard, and Tidewater Minion have to be used to take down the opponent's creatures every turn, as well as white mana from Calciform Pools, and taking one down almost doubles the number of them. The resulting number of turns should be between 2^^42 and 2^^43.
Running the numbers:
T1 Pools
T2 Orchard
opponent's T2 charge Pools (1)
opponent's T3 attack for 1 (19), charge Pools (2)
opponent's T4 attack for 2 (17), charge Pools (3)
opponent's T5 attack for 3 (14), charge Pools (4)
T6 charge Pools (5), empty Pools, Tidewater Minion
opponent's T6 no attack (would have 10 life/4 tokens if they did, but they'd rather keep the tokens)
opponent's T7 no attack (would be 7/3), tap, untap, charge Pools (1)
opponent's T8 no attack (would be 5/2), tap, untap, charge Pools (2)
opponent's T9 no attack (would be 4/1), tap, untap, charge Pools (3)
opponent's T10 tap, untap, charge Pools (4)
... charge Pools (...)
Twhatever: Orchard for R (6 tokens), drain Pools for 5, Intruder Alarm, Ghost-Lit Raider
opponent's Twhatever: still no attack, more starting tokens means more fuel
Twhatever+1: drain Pools for 8, Coat of Arms floating 3, tap Orchard twice using Minion to get GG (8 tokens), Splinterfright
----------
opponent's Twhatever+1: swing with 8 9/9s
Minion/Orchard can of course produce unlimited mana of any color, but the catch is that every single token gets +1/+1 for each mana you make that way. So to pay for Raider's ability with only Orchard mana, they'd get +3/+3 and the target takes 2 damage, a fruitless endeavor. To make progress, only the R can come from Orchard, and the rest must be stored beforehand.
pick off all attackers: 9+17+33+65+129+257+513+1025 red mana, opponent now has 2048 tokens
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Twhatever+2: swing with Splinterfright, and might as well swing with Ghost-Lit Raider too so they have to assign a blocker to it (otherwise it it's obviously lethal with the Coast of Arms Spirit bonus). Won't make a difference now, since you can't afford to lose Raider and it has the same toughness as the blocker's power, but hey.
pick off all blockers, including the one in front of Raider: roughly 2^2059 mana, opponent has 2^2059 tokens after it's too late to block with them.
Ghost-Lit Raider was already blocked so it won't do anything, but Splinterfright gets through for 1 trample damage (19).
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opponent's Twhatever+2: pick off attackers, 2^2^2059 mana, over 2^^5
Twhatever+3: 1 trample damage (18), 2^^6 tokens
...
Twhatever+17: trample damage (4), 2^^34 tokens
Twhatever+18: trample damage (3), 2^^36 tokens
Twhatever+19: trample damage (2), 2^^38 tokens
opponent's Twhatever+19: 2^^39 tokens
----------
Twhatever+20: This time, go ahead and pay 4 to lift the veil of defender off Minion, then alpha strike with all three creatures. Ghost-Lit Raider threatens huge damage by itself, as always, and even Tidewater Minion's 5 damage is something they can't afford to take. So each of them gets a blocker, while Splinterfright faces the rest.
In fact, this is the turn where you can finally ignore the blockers in front of Raider and Minion. Because Pools already had to tap earlier in the turn to get hold of 4 mana, the first Orchard tap here ends up pulling the last of the counters off Pools, rather than paying a R cost for Raider. It still works out better that way, compared to taking out the blocker that's now occupied with Minion.
Once all the Splinterfright blockers are clear, instead of using Ghost-Lit Raider to take down its blocker, it should start damaging itself. This is just as costly as going after a token, so the overall cost is 2^(X-1) rather than 2^(X-2), and with X values that high the difference is of course cosmetic. But when Raider kills itself, Splinterfright gets a +1/+1 bonus from the card in the graveyard, and now deals 2 damage instead of 1, finishing the opponent off.
----------
It costs over 2^^40 mana to achieve this final turn, and 2^^40 turns of charging Pools with itself and Minion alone. That's a bit faster than your original estimate, but comfortably ahead of the Tamiyo/Mercy Killing engine.
The trick where the Raider kills itself on the last turn could be eliminated by replacing Splinterfright with another 0/X Trample. Wilderness Elemental seems to fit the bill, also requiring an extra Orchard tap to cast.
Edit: Unfortunately, I think I see a line that wins quickly without Coat of Arms. After charging up a bit and getting Tidewater Minion, Ghost-Lit Raider, and Intruder Alarm into play, we're safe from attacks because we can kill any number of attacking Spirits, producing one Spirit each. We can then attack with the Minion, and it gets blocked by 6 Spirits. The Raider kills off 3 of the Spirits (producing 3 more), and the Minion kills the other 3. We then safely attack for a few more turns for a quick win.
I don't see an easy fix. We could either replace Raider with something that deals 3+ damage for 1CC or more (a quick search didn't turn up anything. Soul of Shandalar is close, but it hits players. Thornscape Master is 1 damage short), or replace Minion with an X/1 or 0/X (Juniper Order Druid / Stone-Seeder Hierophant) and Wilderness Elemental with a 0/X Trample that shares a type with something (I couldn't find any).
Wilderness Elemental has two colored mana in its cost, just like Splinterfright, and requires two fewer (virtually meaningless) colorless mana. Anything with double-colored mana other than W or U can only be paid for with the help of both Tidewater Minion and Intruder Alarm. The thing is, Wilderness Elemental is a 0/3 by default, so you can play it one turn without fear of dying, then allow them to attack fruitlessly with eight 1/1s the next turn (six will get through, and they won't actually do this). Then you play Coat of Arms the next turn, starting the clock from the turn you put them at 19, instead of their turn immediately prior. That, then, only slows you down by one round of destruction, passing 2^^41 but not 2^^42.
Savage Firecat is 5 mana with double color, just like Splinterfright, and has been given errata to Elemental Cat. But since it starts out at 7/7, it too can be safely played on a board without Coat of Arms (or Ghost-Lit Raider, so they only have 7 tokens when you get it instead of 8). The decision tree is complicated, but suffice it to say quick cat strikes don't actually succeed, and this comes in exactly two turns slower than Wilderness Elemental.
Sutured Ghoul, with its 7 mana and triple color, would be ideal, but one of the other creatures needs to be replaced with a Zombie to allow it to survive. This could be done with Smokespew Invoker over Ghost-Lit Raider, but the fact that it chews through a whopping seven storage counters per activation is unfortunately not as important as the fact that a kill only grows the spirit army by 1.5^n, not 2^n. So after the first incoming attack, instead of facing down 2048 tokens, there are only 304, and the numbers continue to lag way behind.
As far as other pingers that do work at 2^n, Sacellum Archers is a possibility, but it has no type-sharing partner to fit in with. Heedless One is 2/2 without Coat of Arms, which is already enough.
Edit: Unfortunately, I think I see a line that wins quickly without Coat of Arms. After charging up a bit and getting Tidewater Minion, Ghost-Lit Raider, and Intruder Alarm into play, we're safe from attacks because we can kill any number of attacking Spirits, producing one Spirit each. We can then attack with the Minion, and it gets blocked by 6 Spirits. The Raider kills off 3 of the Spirits (producing 3 more), and the Minion kills the other 3. We then safely attack for a few more turns for a quick win.
This sounds all well and good, but what happens after they throw their last few tokens on the altar? Now your Minion is tapped, and thanks to Intruder Alarm, it doesn't untap. The only way you can untap it is by giving them another token, so they will always have chump blockers available if you plan to attack. In fact, they can just single-block it every time and remain in a stable fluctuation between 5 and 6 tokens.
Edit: Unfortunately, I think I see a line that wins quickly without Coat of Arms. After charging up a bit and getting Tidewater Minion, Ghost-Lit Raider, and Intruder Alarm into play, we're safe from attacks because we can kill any number of attacking Spirits, producing one Spirit each. We can then attack with the Minion, and it gets blocked by 6 Spirits. The Raider kills off 3 of the Spirits (producing 3 more), and the Minion kills the other 3. We then safely attack for a few more turns for a quick win.
This sounds all well and good, but what happens after they throw their last few tokens on the altar? Now your Minion is tapped, and thanks to Intruder Alarm, it doesn't untap. The only way you can untap it is by giving them another token, so they will always have chump blockers available if you plan to attack. In fact, they can just single-block it every time and remain in a stable fluctuation between 5 and 6 tokens.
To untap the Minion, tap the Orchard, making a token and floating R. Then tap the Pools to pull off 6 counters, tap the Raider to kill the token, and sink 4 mana into Minion. This allows clear attacks.
Also, I realize now that replacing Minion with a smaller untapper would not fix the deck, as then the opponent could win with the tokens. We need a better pinger if this is going to work.
If you change Ghost-Lit Raider to Sacellum Archers, does it matter that Archers are 2/3 and can survive a chump? Archer cannot pick off tokens until they actually get a chance to attack or block, so at least Minion will have no way to break through.
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Transcendence gets to 2^^18, thanks mostly to the acrobatics that are required to get the damn enchantment safely onto the board without dying. By taking the opponent down to 15 life, they get boosted up to 838,860,802 elves and one spirit. Naturally, this is faster than a 3^^18 turn clock.
My count on the tokens is a little different though. This is what I have for the start:
Their turn: nothing
Your turn: Ping opponent to 19 (2 spirits, 1 elf), tap down spirit.
Their turn: attack with 1 elf, 1 spirit (17 life)
Your turn: Ping opponent to 18 (4 spirits, 1 elf), tap down a spirit.
Their turn: attack with 1 elf, 3 spirits (4 life)
Your turn: Ping opponent to 17 (6 spirits, 1 elf), tap down an elf
Their turn: attack, kill elf then 5 spirits (1 spirit gets through) (1 spirits, 41 elves, 3 life)
Your turn: Ping opponent to 16 (3 spirits, 41 elves), tap down an elf.
Their turn: attack, kill 40 elves then 2 spirits, (1 spirit, ~40*3^40 elves, 2 life)
Here's what I have so far with Transcendence:
Their turn: Kill the spirit (1 elf token)
Your turn: Ping opponent to 19 (1 spirit, 1 elf), tap down spirit.
Their turn: attack with elf, kill the elf (2 spirit, 1 elf)
Your turn: Ping opponent to 18 (2 spirit, 1 elf), tap down a spirit.
Their turn: attack with elf, spirit, kill both (1 spirit, 3 elves)
Your turn: Ping opponent to 17 (2 spirits, 3 elves), tap down an elf
Their turn: attack with 2 spirits, 2 elves, kill all (12 elves)
Your turn: Ping opponent to 16 (1 spirit, 12 elves), tap down an elf
Their turn: attack with 1 spirit, 11 elves, kill all (22,530 elves)
Static Orb might be better than Primal Vigor since it means I can only ping every other turn, but again I haven't done the math on it yet.
-float R with Orchard, give them a spirit (2 cards in hand)
-play Transcendence (1 card in hand)
-respond to the spell with Mercy Killing, targeting the spirit (0 cards in hand)
-respond to both by using the floating mana for Megaliths, targeting self to 19
-Mercy Killing turns spirit into elf
-Tamiyo emblem returns Mercy Killing to hand
-Transcendence resolves
-Tap down the elf (not a spirit)
Except for that one-off deal (and the final turn where you shoot them to 0 right after untapping), you absolutely cannot afford to activate Megaliths on your own turn, since that will necessarily leave all your lands tapped, with no way of drawing more mana on their turn to play Mercy Killing and survive the attack. The following turns look like this:
-on their end step, Mercy Kill their elf, giving back 1 elf, and respond to the Tamiyo trigger with Megaliths to 19, giving them 1 spirit
-your turn, just lock down the spirit
-on their attack, kill their elf and replace it with another elf, then respond with Megaliths to 18 and 2 spirits
-again, lock down a spirit
-their attack, kill elf (1) followed by spirit (2), respond to the last return trigger with Megaliths to 17: 3 elves, 2 spirits
-next turn kill elf (3), elf (5), elf (9), spirit (2), Megaliths to 16: 19 elves, 2 spirits
-kill elf (19; 37; 73; 145; 289; 577; 1153; 2305; 4609; 9217; 18433; 36865; 73729; 147457; 294913; 589825; 1179649; 2359297; 4718593), spirit (2), Megaliths to 15: 9,437,187 elves (> 2^^4), 2 spirits
I was trying to find a way to do without Tamiyo, such as Reiterate + Remand, but those color requirements are crazy and leave nothing that could fit as a win condition.
The fact that Megaliths alone lends itself to a 6-card deck seemed promising for Cyclone along with something like Glimmerpost, but that forces no obligation to drop Coat of Arms (so Cyclone's payments will in fact kill all the tokens that get created). Megaliths are just too versatile to be cleanly replaced by either another land, or another win condition.
Static Orb will get to 2^^29, though, jumping far out into the lead.
Jeeze, Static Orb makes this even more ridiculous. Great job!
If you replace Tamiyo with Reiterate and Calciform Pools with Fungal Reaches does that work out? Unfortunately, you have to find some way to force the Forbidden Orchard to tap, which probably makes it worse in the end since you can't use Primal Vigor or Static Orb.
The trouble with Reiterate by itself is eventually the turn will have to pass, and the original Mercy Killing resolves. When it does, there's no way to bring it back to hand, so you're out of luck for surviving the next strike. We need Remand to get everything back in hand (Reiterate the Remand twice, targeting the non-buyback spells), so that's a commitment to three different colors of mana every turn. Storage lands don't do that.
That looks right--for some reason I had the token count after the first ping listed as 1:1, instead of 1:0, which threw off everything as far as safe attacks. The token count still looks to be a lot more than 900 going against 14, so here's what I have now:
-Tamiyo, Orb, Coat (1 in hand)
-untap, pass turn since there's nothing else productive to do
-opponent's next end step, tap Orchard for R (1 spirit), tap Pools to draw 4 of its counters into mana, Mercy Killing on the spirit (0 in hand), and respond with Megaliths to 19. The spirit gets replaced by an elf.
-untap two lands, lock down the elf
-untap the last land, lock it down again
-end step, Mercy Killing to spin the elf into another elf, and respond with Megaliths to 18 (1 elf, 1 spirit)
-untap 2, lock down the spirit this time
-take 1 from elf, to 19
-untap last land, lock down spirit again
-Mercy Killing to spin the attacking elf into another elf, Megaliths to 17 (1 elf, 2 spirits)
-untap 2, lock a spirit
-take 3 damage to 16
-untap last, lock spirit
-Mercy Killing only once on the attacking spirit, Megaliths to 16, take 2 from attacking elf to 14 (2 elves, 2 spirits)
-untap 2, lock something
-take 6 damage to 8
-untap last, lock elf
-Mercy Killing once on a spirit, Megaliths to 15, take 6 from remaining attackers to 2 (4 elves, 2 spirits)
-untap Megaliths and Orchard, lock spirit
-tap Megaliths to draw from Pools, Mercy Killing five times to survive the attack: 4, 7, 13, 25, 2 (27 elves, 2 spirits)
-untap Megaliths and Pools, lock spirit
-Mercy Killing on all 28 attackers: 27, 53, 105, ..., 1744830465, 2, Megaliths to 14 (3489660929 elves, 2 spirits, > 2^^4)
-2^^5 elves
-2^^6 elves, Megaliths to 13
...
-2^^30 elves, Megaliths to 1
-untap 2 lands
-draw from pools, 2^^31
-untap all lands, Mercy Killing once, respond with Megaliths to 0
That gives me a total of >2^^29 turns.
Numbers in the E273 range are certainly larger than 2^^3, but they're also larger than 2^^4, which is a relatively tame 65536. It's inconvenient to start counting from 2^^5 since the numbers have to be at least E19728, so that's a long way off, and here the final figure works out over 2^^29.
If it cost 2R instead, then it would be simple to drop it and Tamiyo without playing Orchard at all, let both lands get destroyed and returned, and decline the Tamiyo trigger to return Steam Vines. It sits harmlessly in the graveyard for the rest of the game while the opponent has no creatures, and then the real charging phase can begin.
Also this is nuts.
Another idea I had is Quicksilver Fountain, which creates 3-turn periods during which only two can be used to activate Keldon Megaliths' ability; only one of them can be used once life is low enough that Mercy Killing is required every turn. However, if the opponent has enough cards replaced by basic lands, the opponent can play four lands, delaying the unflooding and making white mana unavailable once every four turns, and then there is no way to win.
[Edit addition] Here is another, very different, deck I came up with that also achieves tetration, but not as much as the latest ones: Bottomless Vault; Venser's Journal; Monomania; Rise of the Dark Realms; Derevi, Empyrial Tactician; Krenko, Mob Boss; Sky Swallower.
-Terror is an emotion which, when experienced, results in death.
-The pox was a disease notorious for having killed one-third, rounded up, of Europe’s population. Smallpox, on the other hand, killed only a single person.
-A person riding a horse cannot be stopped by foot soldiers, large animals, walls, archers, or even catapults.
More facts of magic
t1 Megaliths
t2 Pools
t7 Tamiyo
t12 emblem
t13 Orchard, Steam Vines (on Megaliths), give them a single token and lock it down with Tamiyo
t14 tap Megaliths, move Steam Vines onto Pools, tap Pools, drain it of its 5 counters, move Steam Vines onto Orchard, tap Orchard with Tamiyo (they do not get another token from this), use Tamiyo's emblem to get back all three lands but let Steam Vines stay in the graveyard for the rest of the game, replay Megaliths, and finally use the floating mana to play Coat of Arms while you still have the chance
opponent's t14 attack for 1, just take it (go to 16)
t15 Pools, start charging, start locking down the Spirit token again
etc.
and, with 16 life still in reserve at this point, it's just 2^^17.
Forbidden Orchard is used four times to charge Calciform Pools in the opponent's turns and once more to cast Tidewater Minion, which then stops the opponent from winning with the five spirits, and is used to charge Calciform Pools for a long time. Eventually, Ghost-Lit Raider is cast, and on the next turn, Intruder Alarm, Coat of Arms, and Splinterfright are cast. From then on, Ghost-Lit Raider, Forbidden Orchard, and Tidewater Minion have to be used to take down the opponent's creatures every turn, as well as white mana from Calciform Pools, and taking one down almost doubles the number of them. The resulting number of turns should be between 2^^42 and 2^^43.
T1 Pools
T2 Orchard
opponent's T2 charge Pools (1)
opponent's T3 attack for 1 (19), charge Pools (2)
opponent's T4 attack for 2 (17), charge Pools (3)
opponent's T5 attack for 3 (14), charge Pools (4)
T6 charge Pools (5), empty Pools, Tidewater Minion
opponent's T6 no attack (would have 10 life/4 tokens if they did, but they'd rather keep the tokens)
opponent's T7 no attack (would be 7/3), tap, untap, charge Pools (1)
opponent's T8 no attack (would be 5/2), tap, untap, charge Pools (2)
opponent's T9 no attack (would be 4/1), tap, untap, charge Pools (3)
opponent's T10 tap, untap, charge Pools (4)
... charge Pools (...)
Twhatever: Orchard for R (6 tokens), drain Pools for 5, Intruder Alarm, Ghost-Lit Raider
opponent's Twhatever: still no attack, more starting tokens means more fuel
Twhatever+1: drain Pools for 8, Coat of Arms floating 3, tap Orchard twice using Minion to get GG (8 tokens), Splinterfright
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opponent's Twhatever+1: swing with 8 9/9s
Minion/Orchard can of course produce unlimited mana of any color, but the catch is that every single token gets +1/+1 for each mana you make that way. So to pay for Raider's ability with only Orchard mana, they'd get +3/+3 and the target takes 2 damage, a fruitless endeavor. To make progress, only the R can come from Orchard, and the rest must be stored beforehand.
pick off all attackers: 9+17+33+65+129+257+513+1025 red mana, opponent now has 2048 tokens
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Twhatever+2: swing with Splinterfright, and might as well swing with Ghost-Lit Raider too so they have to assign a blocker to it (otherwise it it's obviously lethal with the Coast of Arms Spirit bonus). Won't make a difference now, since you can't afford to lose Raider and it has the same toughness as the blocker's power, but hey.
pick off all blockers, including the one in front of Raider: roughly 2^2059 mana, opponent has 2^2059 tokens after it's too late to block with them.
Ghost-Lit Raider was already blocked so it won't do anything, but Splinterfright gets through for 1 trample damage (19).
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opponent's Twhatever+2: pick off attackers, 2^2^2059 mana, over 2^^5
Twhatever+3: 1 trample damage (18), 2^^6 tokens
...
Twhatever+17: trample damage (4), 2^^34 tokens
Twhatever+18: trample damage (3), 2^^36 tokens
Twhatever+19: trample damage (2), 2^^38 tokens
opponent's Twhatever+19: 2^^39 tokens
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Twhatever+20: This time, go ahead and pay 4 to lift the veil of defender off Minion, then alpha strike with all three creatures. Ghost-Lit Raider threatens huge damage by itself, as always, and even Tidewater Minion's 5 damage is something they can't afford to take. So each of them gets a blocker, while Splinterfright faces the rest.
In fact, this is the turn where you can finally ignore the blockers in front of Raider and Minion. Because Pools already had to tap earlier in the turn to get hold of 4 mana, the first Orchard tap here ends up pulling the last of the counters off Pools, rather than paying a R cost for Raider. It still works out better that way, compared to taking out the blocker that's now occupied with Minion.
Once all the Splinterfright blockers are clear, instead of using Ghost-Lit Raider to take down its blocker, it should start damaging itself. This is just as costly as going after a token, so the overall cost is 2^(X-1) rather than 2^(X-2), and with X values that high the difference is of course cosmetic. But when Raider kills itself, Splinterfright gets a +1/+1 bonus from the card in the graveyard, and now deals 2 damage instead of 1, finishing the opponent off.
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It costs over 2^^40 mana to achieve this final turn, and 2^^40 turns of charging Pools with itself and Minion alone. That's a bit faster than your original estimate, but comfortably ahead of the Tamiyo/Mercy Killing engine.
The trick where the Raider kills itself on the last turn could be eliminated by replacing Splinterfright with another 0/X Trample. Wilderness Elemental seems to fit the bill, also requiring an extra Orchard tap to cast.
Edit: Unfortunately, I think I see a line that wins quickly without Coat of Arms. After charging up a bit and getting Tidewater Minion, Ghost-Lit Raider, and Intruder Alarm into play, we're safe from attacks because we can kill any number of attacking Spirits, producing one Spirit each. We can then attack with the Minion, and it gets blocked by 6 Spirits. The Raider kills off 3 of the Spirits (producing 3 more), and the Minion kills the other 3. We then safely attack for a few more turns for a quick win.
I don't see an easy fix. We could either replace Raider with something that deals 3+ damage for 1CC or more (a quick search didn't turn up anything. Soul of Shandalar is close, but it hits players. Thornscape Master is 1 damage short), or replace Minion with an X/1 or 0/X (Juniper Order Druid / Stone-Seeder Hierophant) and Wilderness Elemental with a 0/X Trample that shares a type with something (I couldn't find any).
Savage Firecat is 5 mana with double color, just like Splinterfright, and has been given errata to Elemental Cat. But since it starts out at 7/7, it too can be safely played on a board without Coat of Arms (or Ghost-Lit Raider, so they only have 7 tokens when you get it instead of 8). The decision tree is complicated, but suffice it to say quick cat strikes don't actually succeed, and this comes in exactly two turns slower than Wilderness Elemental.
Sutured Ghoul, with its 7 mana and triple color, would be ideal, but one of the other creatures needs to be replaced with a Zombie to allow it to survive. This could be done with Smokespew Invoker over Ghost-Lit Raider, but the fact that it chews through a whopping seven storage counters per activation is unfortunately not as important as the fact that a kill only grows the spirit army by 1.5^n, not 2^n. So after the first incoming attack, instead of facing down 2048 tokens, there are only 304, and the numbers continue to lag way behind.
As far as other pingers that do work at 2^n, Sacellum Archers is a possibility, but it has no type-sharing partner to fit in with. Heedless One is 2/2 without Coat of Arms, which is already enough.
This sounds all well and good, but what happens after they throw their last few tokens on the altar? Now your Minion is tapped, and thanks to Intruder Alarm, it doesn't untap. The only way you can untap it is by giving them another token, so they will always have chump blockers available if you plan to attack. In fact, they can just single-block it every time and remain in a stable fluctuation between 5 and 6 tokens.
To untap the Minion, tap the Orchard, making a token and floating R. Then tap the Pools to pull off 6 counters, tap the Raider to kill the token, and sink 4 mana into Minion. This allows clear attacks.
Also, I realize now that replacing Minion with a smaller untapper would not fix the deck, as then the opponent could win with the tokens. We need a better pinger if this is going to work.