Start by casting Fathom Mage, then cast Saw in Half on it to make two of them (still 1/1). Now cast Keeper of the Cadence. It triggers evolve on both Fathom Mages. Activate Keeper of the Cadence to put Saw in Half into the library, then resolve the first evolve and the on-evolve trigger to draw it. Cast it on the now-2/2 Fathom Mage, making two more 1/1 Fathom Mages, then resolve the rest, redrawing Saw in Half again. (We now have one 2/2 and two 1/1 Fathom Mages.)
Cast Saw in Half on Keeper of the Cadence, making two 1/3s, which both trigger evolve on the Fathom Mages. Let the 2/2 Fathom Mage evolve to 3/3 first, and Saw it into two 2/2s. We get four more redraws of Saw from evolving the two 1/1 Fathom Mages to 3/3; use the first three to Saw the new 2/2 Fathom Mages into four 1/1s, and one of those into two 1/1s. (We now have two 3/3 Fathom Mages, five 1/1 Fathom Mages, and two 1/3 Keeper of the Cadence, and Saw in Half in hand.)
The process used now is: Saw a ?/3 into two ?/2s, triggering evolve on all the 1/1 Fathom Mages. Each time a 1/1 Fathom Mage evolves into 2/2, it redraws Saw, which is cast on that now-2/2 to turn it into two 1/1 Fathom Mages; except for the last time, where we keep Saw in hand. The net result is to go from X 1/1 Fathom Mages to (X-1)*2 1/1 Fathom Mages and one 2/2 Fathom Mage (as well as whatever was produced by the initial Saw).
Follow that process using the two 3/3 Fathom Mages and one 1/3 Keeper of the Cadence. We now have 26 1/1 Fathom Mages, 7 2/2 Fathom Mages, two 1/2 Keeper of the Cadence, and one 1/3 Keeper of the Cadence.
Cast Life of the Party. Cast Saw in Half on the remaining 1/3 Keeper of the Cadence, triggering evolve on the 26 1/1 Fathom Mages. Use the first 25 draws to Saw Life of the Party, ending with 26 Life of the Party. Attack with them.
Note that this arrangement has some losses at the higher values, because of evolve only increasing power and toughness by one at a time. However, the hyperoperations should quickly overwhelm those losses.
Unfortunately, adding Black Lotus, Channel and Lich's Mirror for the no-mana-given version doesn't work with these cards, because it doesn't provide infinite mana during combat.
Notes for potential modifications: Sharp-Eyed Rookie can replace Fathom Mage. Watchful Radstag seems potentially useful but is redundant if we're using Saw in Half.
For 8 cards, the latest write-up specifies "stacking the triggers to delay the ones that would turn larger Shapeshifter-Golems into smaller ones", but that doesn't work: pretty soon, one is forced to resolve those triggers in order to get to Inverter of Truth reshuffle triggers deeper in the stack.
The correct approach is the opposite: it is the high-value Shapeshifter triggers that should be left on the stack until they are ready to be used. However, it turns out to be more tricky than I previously imagined it would be.
When having a high-power Shapeshifter-Golem to break down, first cast Saw in Half on a Shapeshifter-Inverter. Place the resulting Inverter reshuffle triggers above the Shapeshifter-changing triggers. Resolve one, getting back the Saw in Half, and cast it on the high-power Shapeshifter-Golem. This produces a bunch more Shapeshifter-changing triggers; let them resolve, turning all the Shapeshifters into the halved-power Shapeshifter-Golem, and then resolve the second Inverter reshuffle trigger to get back the Saw in Half. Now here's the critical part: make sure that the next Shapeshifter-Golem to be Sawed is one of the tokens produced from Sawing that first Shapeshifter-Inverter – there are triggers still on the stack to change Shapeshifters into this creature, and Sawing it locks in its last-known information as the halved-power Shapeshifter-Golem.
(If there is no Shapeshifter-Inverter to begin with, we can get one by Sawing a plain Inverter token, and resolving one Inverter reshuffle trigger and one Shapeshifter-changing trigger; we can afford the extra 3 life at this stage.)
How does comeuppance work with multiple arcbonds again? It doesnt say instead, so the damage half is a trigger? If it triggers and we can resolve another arcbond first, can't we just ignore it and do the same multiple arcbond tricks from before?
It's not a triggered ability. (Triggered abilities always use "When", "Whenever", or "At".) It's part of the prevention effect; the damage is also dealt as part of the Arcbond ability resolving.
615.5. Some prevention effects also include an additional effect, which may refer to the amount of damage that was prevented. The prevention takes place at the time the original event would have happened; the rest of the effect takes place immediately afterward.
For 4 cards with unlimited mana, we can do Mondrak, Glory Dominus, Ratadrabik of Urborg, Cadric, Soul Kindler, and... something. Junji, the Midnight Sky would be ideal if not for the life loss. The best I could find is Colfenor, the Last Yew, which adds 2 layers (because the stack has to clear to recast cards) to the 4 from the other cards for a total of 6. (How to do it: Sacrificing each Colfenor token gets multiple triggers to return all the other cards. Cast Ratadrabik and Cadric first and get copies of them. Then, cast Mondrak and get a bunch of Cadric triggers. After each of those triggers resolves and gives a bunch of legendary Mondrak tokens, sacrifice a single legendary Ratadrabik token to get a bunch of Ratadrabik/Ratadrabik triggers, and alternate resolving those triggers with sacrificing legendary Mondrak tokens, so that the Ratadrabik count is boosted for each sacrifice of a legendary Mondrak token.)
Unfortunately, adding Black Lotus, Channel and Lich's Mirror for the no-mana-given version doesn't work with these cards, because it is possible to set off the Lich's Mirror reset with Ratadrabik triggers still on the stack, giving a higher starting point, and repeat to go infinite. If we weren't constrained to Vintage legality, 2x Black Lotus and Auriok Salvagers would work as an alternative way of getting infinite mana.
For 5 cards with unlimited mana, I think we can add Kodama of the East Tree to Mondrak, Glory Dominus, Ratadrabik of Urborg, Cadric, Soul Kindler, and Colfenor, the Last Yew. Ratadrabik can now be the starting point; the Mondrak count doesn't get increased at that point, but we still get a multiplication for each legendary Ratadrabik token, and then an exponentiation for each sacrifice of a Mondrak token getting back Ratadrabik and Cadric, for a layer-1 foundation. After that, Mondrak, Kodama, and Colfenor each add 4 layers above that, for a total of 13 layers.
(Correction) Mondrak doesn't get re-increased enough to count for a layer from multiplying its own nonlegendary tokens, dropping it to 3 layers; thus Mondrak ends up at layer 4 whether you use this procedure or the other procedure from above, and the total is 12 layers.
For 3 cards with unlimited mana: Mondrak, Glory Dominus, Ratadrabik of Urborg, Twinflame making two copies of each; have the originals and one copy of each die to the legend rule, generating six Ratadrabik triggers for Mondrak and four Ratadrabik triggers for Ratadrabik. Resolve the triggers for Mondrak first, getting >2^^6 nonlegendary copies of Mondrak, then resolve the rest for >2^^7 nonlegendary copies of Ratadrabik. Now activate one of the Mondrak tokens, sacrificing the two remaining legendary tokens and getting >2^^7 Ratadrabik triggers for each. Again, resolve the triggers for Mondrak first for >2^^2^^7 copies of Mondrak, and then resolve the rest for >2^^2^^7 copies of Ratadrabik. Those tokens have haste and can attack.
(Cadric, Soul Kindler would have been better if its haste were copiable.)
The total mana consumption is 16, so this can also become a 6-card solution with Black Lotus, Channel, and Chromatic Orrery.
Speaking of alternatives, a recent Reddit post has gotten me thinking about another possible challenge. This one isn't really focusing on cards as a limit, but the restriction is big enough that this thread feels to me like it might be a better place than the 60-card one.
The original challenge was finding the best way to kill the opponent with a Legacy-legal deck that isn't able to cast spells. The original post suggested a turn 3 kill with Dark Depths, and it looks like a dredge strategy can win on turn 1.
So, to build off that, how much damage could we deal? (Perhaps switching from the Legacy card pool to Vintage, but it might not even matter.) One thing that could be helpful is Priest of Fell Rites, since it can get back arbitrary creatures without casting a spell. Angel of Glory's Rise can turn one reanimation into many, and reanimating creatures can turn into reanimating artifacts and enchantments.
The difference between Legacy and Vintage is actually significant for this. In Vintage, we can get started relatively easily: reveal four Chancellor of the Tangle, play Bazaar of Baghdad and activate it and discard two Chancellors and a Golgari Grave-Troll, cycle Street Wraith and dredge the Golgari Grave-Troll, exile a Simian Spirit Guide for one more mana, exile two Jack-o-Lantern from the graveyard to fix mana, and unearth a Priest of Fell Rites with three remaining non-fixed cards in the graveyard and one in hand (or four in graveyard by discarding one instead of a Chancellor).
In Legacy, it's harder to get started. The Geier Reach Sanitarium method you mentioned in the linked post doesn't leave enough cards to get the mana to unearth a Priest of Fell Rites. Starting with Dakmor Salvage + cycle Edge of Autumn does a bit better; continuing similarly to above does allow unearthing a Priest of Fell Rites on the draw only, with four remaining non-fixed cards in the graveyard. But to do it on the play, I needed to use a more complicated method. Start by revealing two Chancellor of the Tangle and putting three Leyline of Anticipation onto the battlefield. Play Nykthos, Shrine to Nyx and activate it for 6 blue mana. Use the last card -- Waker of Waves -- to put a Golgari Grave-Troll in the graveyard and a Vizier of Tumbling Sands in hand. Cycle that, untapping Nykthos, activate Nykthos again for another 6 blue mana, and dredge the Golgari Grave-Troll with the draw. Then, as before, exile two Jack-o-Lantern from the graveyard to fix mana, and unearth a Priest of Fell Rites with three remaining non-fixed cards in the graveyard.
I still count only two Saws for each Shapeshifter ability that is around when we cast the golem. And both of those Saws destroy 1 and create 2 creatures with the ability, whether it's an inverter or golem. I still don't see how you get the *3.
That makes it a change of +2; to get the final number, the change has to be added to the initial number.
Cast a Saw pair on a Shapeshifter-Inverter and the original Golem.
Cast Boulderbranch Golem. (net -1 life) All Shapeshifters trigger.
After each of those Shapeshifter triggers resolve, cast a Saw pair on an Inverter and the 6/5 Shapeshifter-Golem. The first two times, it has to target a plain Inverter; each subsequent time can target a Shapeshifter-Inverter that does not have a waiting trigger to become a 6/5 Golem.
The net result is -1 life, and +1 and then *3-2 to the number of Shapeshifters, which works out to *3+1.
This can be repeated 7 times, ending with 7 life and 9841 Shapeshifters.
Are those numbers correct? The initial saw pair accounts for the +1, but after that we only cast one saw pair for each shapeshifter with a trigger. Each saw increases the number of creatures with the shapeshifter ability by -1+2 = 1. So I think we go from N to (N+1)*2-2 = 2*N instead of the N*3+1 you have.
What you're missing is that the Shapeshifter count is increased both by Sawing Shapeshifter-Golems and by Sawing Shapeshifter-Inverters.
I think I can now exceed Graham's Number in 8 cards!
The main improvement is that Renegade Doppelganger or Unstable Shapeshifter can serve the purpose of Doubling Season, without needing an extra card (which was Astral Dragon). The downside of this is lower initial growth rates; I brought the growth rate back up a bit with precise card selection.
Start with Black Lotus and Show and Tell to put Bolas's Citadel onto the battlefield.
Cast Boulderbranch Golem from library. (-7, +6; 19 life)
Cast Saw in Half from library, targeting Boulderbranch Golem. (-3, +6; 22 life)
Cast Inverter of Truth from library; it moves Black Lotus, Show and Tell, Boulderbranch Golem, and Saw in Half from graveyard into library. (-4; 18 life)
Cast Black Lotus, and cast Boulderbranch Golem. (-7, +6; 17 life)
Cast Saw in Half on Inverter of Truth. Two copies trigger. (-3; 14 life)
The first trigger exiles Show and Tell and sends Saw in Half and Inverter of Truth into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting Boulderbranch Golem. (-3, +6; 17 life)
The second trigger exiles Inverter of Truth and sends Black Lotus, Saw in Half, and Boulderbranch Golem into the library.
Cast Black Lotus, and cast Boulderbranch Golem. (-7, +6; 16 life)
Cast Saw in Half on an Inverter of Truth. Two copies trigger. (-3; 13 life)
The first trigger sends Saw in Half into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting Boulderbranch Golem. (-3, +6; 16 life)
The second trigger sends Black Lotus, Saw in Half, and Boulderbranch Golem into the library.
With enough mana now, cast Unstable Shapeshifter from hand, and then cast Black Lotus and Boulderbranch Golem (-7, +6; 15 life); Unstable Shapeshifter becomes a copy of Boulderbranch Golem.
Cast Saw in Half on an Inverter of Truth. (-3; 12 life) Two copies trigger, and Unstable Shapeshifter triggers twice; put the Inverter triggers on top.
The first Inverter trigger sends Saw in Half into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting the Shapeshifter-Golem. (-3, +6; 15 life)
The second trigger sends Black Lotus, Saw in Half, and Unstable Shapeshifter into the library.
Cast Black Lotus one more time, then cast another pair of Saws on an Inverter and the original Golem (life-neutral). (Unstable Shapeshifter gets exiled.)
Cast Boulderbranch Golem. (-7, +6; 14 life) Both Shapeshifters trigger to become the full 6/5 Golem.
Let those triggers resolve one by one, and after each one resolves, cast another life-neutral Saw pair on an Inverter and the 6/5 Shapeshifter-Golem. There are now four Unstable Shapeshifters, two copying 3/3 Golems and two copying Inverters.
Now we can begin the full loop:
Cast a Saw pair on a Shapeshifter-Inverter and the original Golem.
Cast Boulderbranch Golem. (net -1 life) All Shapeshifters trigger.
After each of those Shapeshifter triggers resolve, cast a Saw pair on an Inverter and the 6/5 Shapeshifter-Golem. The first two times, it has to target a plain Inverter; each subsequent time can target a Shapeshifter-Inverter that does not have a waiting trigger to become a 6/5 Golem.
The net result is -1 life, and +1 and then *3-2 to the number of Shapeshifters, which works out to *3+1.
This can be repeated 7 times, ending with 7 life and 9841 Shapeshifters.
Cast the final card, Life of the Party, from hand. All the Shapeshifters turn into that.
Attack with all possible creatures, getting triggers from Life of the Party and the Shapeshifter copies of it; put the original's trigger on the bottom.
Cast a Saw pair on an Inverter and the original Golem, this time letting the triggers turning Shapeshifters into Inverters resolve first, so that afterwards they become Golems and get to stay Golems for a while.
A trigger from Life of the Party resolves, giving a big boost to the power of a Shapeshifter that is now copying a Golem. Start Sawing again with that one.
(As before, leave the high-value Shapeshifter triggers on the stack until they are ready to be consumed. Now that there is excess life being gained from the big Golems, it can be spent on extra Saws on small Golems. This should work out to a number of Knuth arrows somewhere around the base-2 logarithm of the starting power.)
Repeat with each Life of the Party trigger, until the final one resolves for the original Life of the Party, which is left to deal the big damage.
Queue of Beetles and Red Herring, from the Mystery Booster Playtest Cards, are the only cards I could find that modify the stack in any way other than the usual possibilities of removing items and adding items on top.
There are also a few cards that can remove a spell from the stack and put it back on, which is similar to moving it to the top: Narset's Reversal, Psychic Rebuttal, and Guile.
One other possible point of variation is defining exactly what it means to be limited to N cards. That could mean starting with a deck of only N cards and having an otherwise-empty library, or assuming that there's other cards filling space there and we just can't use them. This affects the power of cards like Ardent Dustspeaker and Junktroller that put cards on the bottom of the library.
That can be improved slightly by replacing Omniscience with Aluren, which allows casting at instant speed. This means we can venture through Lost Mine of Phandelver one extra time at the beginning, drawing Devilish Valet, while leaving the token-creating trigger from Goblin Lair on the stack, for one extra trigger of Devilish Valet. It also means we can leave Devilish Valet's triggers on the stack and resolve all the Storeroom triggers first. This increases the final power to 56*2^284.
Another benefit of Aluren is that it costs only 4 mana, but I couldn't find any way to get a big advantage from that; the best I have is starting with Black Lotus and Sunscorched Desert for 1 extra damage. (Crystal Dragon looked promising, but there's no way to get the required mana to cast it from exile without giving up multiple Goblin tokens.)
For 2 cards, Devilish Valet and Displacer Beast is better. All three default dungeons have unavoidable limiting effects: "Each player loses 1 life" in Tomb of Annihilation, and card draw at the end of Lost Mine of Phandelver and Dungeon of the Mad Mage.
Start by casting Displacer Beast four times to venture through Lost Mine of Phandelver, getting a Goblin token at Goblin Lair, proceeding from there into Storeroom, and drawing Devilish Valet at the end. Cast Devilish Valet. Then go through Tomb of Annihilation 19 times. At Oubliette, sacrifice the Goblin token the first time, and sacrifice The Atropal on later times. This triggers Devilish Valet 19*4=76 times. Finally, venture six steps into Dungeon of the Mad Mage, going through Muiral's Graveyard, for 8 more triggers of DV, making its power 2^84, and attack.
5 cards: Black Lotus, Channel, Dryad's Revival retrieving Black Lotus, use it for green mana, flashback Dryad's Revival retrieving Black Lotus, use it for red mana, Nylea's Colossus, Rionya, Fire Dancer. Go to combat and make 4 hasty copies of Nylea's Colossus, producing 20 of its triggers, all targeting one of the copies. Attack for 6*2^20+18=6291474 damage.
One permanent can be sacrificed to bounce Kaya, Geist Hunter and double the current token multiplier.
One Whale produces many Krakens, for an exponentiation.
One Fish produces many Whales, for a tetration.
One Reef Worm produces many Fishes, for a pentation.
Exiling Reef Worm to Dollhouse of Horrors produces many of them, for a hexation.
When the original Reef Worm is sacrificed, it produces two Fishes; after those are fully consumed, the token multiplier is >2^^^4. Then Dollhouse of Horrors makes >2^^^4 copies of Reef Worm, for an end result of >2^^^^(2^^^4). Hold on to one of the Reef Worm tokens, and after the last Whale is sacrificed, start using the first ability of Kaya, Geist Hunter to put +1/+1 counters on it, so that it can attack for >2^^^^(2^^^4) damage.
Unfortunately, this can't be made into a 7-card solution by adding Black Lotus, Channel, and Lich's Mirror, because the token-doubling effects persist through Lich's Mirror's effect.
1st LF copy gives 6 life and one Golem is destroyed. Activate CA targeting PG.
2nd LF copy gives 8 life and one Golem is destroyed. Activate CA targeting PG. 1 mana left over.
3rd LF copy gives 10 life and one Golem is destroyed. Activate CA targeting PG. 4 mana left over.
4th LF copy gives 12 life and one Golem is destroyed. Activate CA targeting PG twice. 2 mana left over.
5th LF copy gives 17 life and one Golem is destroyed. Activate CA targeting PG twice. 5 mana left over.
Second PG trigger copies LF 21 times.
etc.
For the final card, like before, we want an instant/sorcery that can target a Golem and draw a card (or otherwise obtain Lantern Flare from the library to be recast), but that's not all this time. The Golems are being destroyed because of Lantern Flare's damage, which normally makes the final spell fizzle. We want to choose a card that can avoid fizzling.
Oblivion's Hunger would work if not for the +1/+1-counter condition.
Shelter would work if it were a different color from Lantern Flare. (Harsh Sustenance in place of Lantern Flare would solve this problem, but its mana requirements make it impossible to get going.)
Survive the Night would work, except that it goes infinite by saving a Clue to redraw it after it resolves, or by using Cogwork Assembler to copy a Clue.
Adrix and Nev, Twincasters, Ratadrabik of Urborg, Cabaretti Confluence... but the haste isn't copiable. If it were, we could make 2 copies of A&N, make 8 copies of A&N, and then make 2048 copies of Ratadrabik, and get 20490 Ratadrabik triggers on A&N and a bunch on Ratadrabik, and deal >2^^20492 damage.
Instead, Kindred Charge is the best I could find, dealing >2^^8 damage.
Start by casting Fathom Mage, then cast Saw in Half on it to make two of them (still 1/1). Now cast Keeper of the Cadence. It triggers evolve on both Fathom Mages. Activate Keeper of the Cadence to put Saw in Half into the library, then resolve the first evolve and the on-evolve trigger to draw it. Cast it on the now-2/2 Fathom Mage, making two more 1/1 Fathom Mages, then resolve the rest, redrawing Saw in Half again. (We now have one 2/2 and two 1/1 Fathom Mages.)
Cast Saw in Half on Keeper of the Cadence, making two 1/3s, which both trigger evolve on the Fathom Mages. Let the 2/2 Fathom Mage evolve to 3/3 first, and Saw it into two 2/2s. We get four more redraws of Saw from evolving the two 1/1 Fathom Mages to 3/3; use the first three to Saw the new 2/2 Fathom Mages into four 1/1s, and one of those into two 1/1s. (We now have two 3/3 Fathom Mages, five 1/1 Fathom Mages, and two 1/3 Keeper of the Cadence, and Saw in Half in hand.)
The process used now is: Saw a ?/3 into two ?/2s, triggering evolve on all the 1/1 Fathom Mages. Each time a 1/1 Fathom Mage evolves into 2/2, it redraws Saw, which is cast on that now-2/2 to turn it into two 1/1 Fathom Mages; except for the last time, where we keep Saw in hand. The net result is to go from X 1/1 Fathom Mages to (X-1)*2 1/1 Fathom Mages and one 2/2 Fathom Mage (as well as whatever was produced by the initial Saw).
Follow that process using the two 3/3 Fathom Mages and one 1/3 Keeper of the Cadence. We now have 26 1/1 Fathom Mages, 7 2/2 Fathom Mages, two 1/2 Keeper of the Cadence, and one 1/3 Keeper of the Cadence.
Cast Life of the Party. Cast Saw in Half on the remaining 1/3 Keeper of the Cadence, triggering evolve on the 26 1/1 Fathom Mages. Use the first 25 draws to Saw Life of the Party, ending with 26 Life of the Party. Attack with them.
Note that this arrangement has some losses at the higher values, because of evolve only increasing power and toughness by one at a time. However, the hyperoperations should quickly overwhelm those losses.
Unfortunately, adding Black Lotus, Channel and Lich's Mirror for the no-mana-given version doesn't work with these cards, because it doesn't provide infinite mana during combat.
Notes for potential modifications: Sharp-Eyed Rookie can replace Fathom Mage. Watchful Radstag seems potentially useful but is redundant if we're using Saw in Half.
The correct approach is the opposite: it is the high-value Shapeshifter triggers that should be left on the stack until they are ready to be used. However, it turns out to be more tricky than I previously imagined it would be.
When having a high-power Shapeshifter-Golem to break down, first cast Saw in Half on a Shapeshifter-Inverter. Place the resulting Inverter reshuffle triggers above the Shapeshifter-changing triggers. Resolve one, getting back the Saw in Half, and cast it on the high-power Shapeshifter-Golem. This produces a bunch more Shapeshifter-changing triggers; let them resolve, turning all the Shapeshifters into the halved-power Shapeshifter-Golem, and then resolve the second Inverter reshuffle trigger to get back the Saw in Half. Now here's the critical part: make sure that the next Shapeshifter-Golem to be Sawed is one of the tokens produced from Sawing that first Shapeshifter-Inverter – there are triggers still on the stack to change Shapeshifters into this creature, and Sawing it locks in its last-known information as the halved-power Shapeshifter-Golem.
(If there is no Shapeshifter-Inverter to begin with, we can get one by Sawing a plain Inverter token, and resolving one Inverter reshuffle trigger and one Shapeshifter-changing trigger; we can afford the extra 3 life at this stage.)
It's not a triggered ability. (Triggered abilities always use "When", "Whenever", or "At".) It's part of the prevention effect; the damage is also dealt as part of the Arcbond ability resolving.
615.5. Some prevention effects also include an additional effect, which may refer to the amount of damage that was prevented. The prevention takes place at the time the original event would have happened; the rest of the effect takes place immediately afterward.
Unfortunately, adding Black Lotus, Channel and Lich's Mirror for the no-mana-given version doesn't work with these cards, because it is possible to set off the Lich's Mirror reset with Ratadrabik triggers still on the stack, giving a higher starting point, and repeat to go infinite. If we weren't constrained to Vintage legality, 2x Black Lotus and Auriok Salvagers would work as an alternative way of getting infinite mana.
For 5 cards with unlimited mana, I think we can add Kodama of the East Tree to Mondrak, Glory Dominus, Ratadrabik of Urborg, Cadric, Soul Kindler, and Colfenor, the Last Yew. Ratadrabik can now be the starting point; the Mondrak count doesn't get increased at that point, but we still get a multiplication for each legendary Ratadrabik token, and then an exponentiation for each sacrifice of a Mondrak token getting back Ratadrabik and Cadric, for a layer-1 foundation.
After that, Mondrak, Kodama, and Colfenor each add 4 layers above that, for a total of 13 layers.(Correction) Mondrak doesn't get re-increased enough to count for a layer from multiplying its own nonlegendary tokens, dropping it to 3 layers; thus Mondrak ends up at layer 4 whether you use this procedure or the other procedure from above, and the total is 12 layers.
For 3 cards with unlimited mana: Mondrak, Glory Dominus, Ratadrabik of Urborg, Twinflame making two copies of each; have the originals and one copy of each die to the legend rule, generating six Ratadrabik triggers for Mondrak and four Ratadrabik triggers for Ratadrabik. Resolve the triggers for Mondrak first, getting >2^^6 nonlegendary copies of Mondrak, then resolve the rest for >2^^7 nonlegendary copies of Ratadrabik. Now activate one of the Mondrak tokens, sacrificing the two remaining legendary tokens and getting >2^^7 Ratadrabik triggers for each. Again, resolve the triggers for Mondrak first for >2^^2^^7 copies of Mondrak, and then resolve the rest for >2^^2^^7 copies of Ratadrabik. Those tokens have haste and can attack.
(Cadric, Soul Kindler would have been better if its haste were copiable.)
The total mana consumption is 16, so this can also become a 6-card solution with Black Lotus, Channel, and Chromatic Orrery.
I have done some looking into that challenge.
The difference between Legacy and Vintage is actually significant for this. In Vintage, we can get started relatively easily: reveal four Chancellor of the Tangle, play Bazaar of Baghdad and activate it and discard two Chancellors and a Golgari Grave-Troll, cycle Street Wraith and dredge the Golgari Grave-Troll, exile a Simian Spirit Guide for one more mana, exile two Jack-o-Lantern from the graveyard to fix mana, and unearth a Priest of Fell Rites with three remaining non-fixed cards in the graveyard and one in hand (or four in graveyard by discarding one instead of a Chancellor).
In Legacy, it's harder to get started. The Geier Reach Sanitarium method you mentioned in the linked post doesn't leave enough cards to get the mana to unearth a Priest of Fell Rites. Starting with Dakmor Salvage + cycle Edge of Autumn does a bit better; continuing similarly to above does allow unearthing a Priest of Fell Rites on the draw only, with four remaining non-fixed cards in the graveyard. But to do it on the play, I needed to use a more complicated method. Start by revealing two Chancellor of the Tangle and putting three Leyline of Anticipation onto the battlefield. Play Nykthos, Shrine to Nyx and activate it for 6 blue mana. Use the last card -- Waker of Waves -- to put a Golgari Grave-Troll in the graveyard and a Vizier of Tumbling Sands in hand. Cycle that, untapping Nykthos, activate Nykthos again for another 6 blue mana, and dredge the Golgari Grave-Troll with the draw. Then, as before, exile two Jack-o-Lantern from the graveyard to fix mana, and unearth a Priest of Fell Rites with three remaining non-fixed cards in the graveyard.
We're not. Boulderbranch Golem is only cast after each batch of Shapeshifter triggers is finished, and is no longer cast once we attack.
The main improvement is that Renegade Doppelganger or Unstable Shapeshifter can serve the purpose of Doubling Season, without needing an extra card (which was Astral Dragon). The downside of this is lower initial growth rates; I brought the growth rate back up a bit with precise card selection.
Start with Black Lotus and Show and Tell to put Bolas's Citadel onto the battlefield.
Cast Boulderbranch Golem from library. (-7, +6; 19 life)
Cast Saw in Half from library, targeting Boulderbranch Golem. (-3, +6; 22 life)
Cast Inverter of Truth from library; it moves Black Lotus, Show and Tell, Boulderbranch Golem, and Saw in Half from graveyard into library. (-4; 18 life)
Cast Black Lotus, and cast Boulderbranch Golem. (-7, +6; 17 life)
Cast Saw in Half on Inverter of Truth. Two copies trigger. (-3; 14 life)
The first trigger exiles Show and Tell and sends Saw in Half and Inverter of Truth into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting Boulderbranch Golem. (-3, +6; 17 life)
The second trigger exiles Inverter of Truth and sends Black Lotus, Saw in Half, and Boulderbranch Golem into the library.
Cast Black Lotus, and cast Boulderbranch Golem. (-7, +6; 16 life)
Cast Saw in Half on an Inverter of Truth. Two copies trigger. (-3; 13 life)
The first trigger sends Saw in Half into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting Boulderbranch Golem. (-3, +6; 16 life)
The second trigger sends Black Lotus, Saw in Half, and Boulderbranch Golem into the library.
With enough mana now, cast Unstable Shapeshifter from hand, and then cast Black Lotus and Boulderbranch Golem (-7, +6; 15 life); Unstable Shapeshifter becomes a copy of Boulderbranch Golem.
Cast Saw in Half on an Inverter of Truth. (-3; 12 life) Two copies trigger, and Unstable Shapeshifter triggers twice; put the Inverter triggers on top.
The first Inverter trigger sends Saw in Half into the library.
Before the second trigger resolves, sacrifice Black Lotus for 3 mana, and cast Saw in Half targeting the Shapeshifter-Golem. (-3, +6; 15 life)
The second trigger sends Black Lotus, Saw in Half, and Unstable Shapeshifter into the library.
Cast Black Lotus one more time, then cast another pair of Saws on an Inverter and the original Golem (life-neutral). (Unstable Shapeshifter gets exiled.)
Cast Boulderbranch Golem. (-7, +6; 14 life) Both Shapeshifters trigger to become the full 6/5 Golem.
Let those triggers resolve one by one, and after each one resolves, cast another life-neutral Saw pair on an Inverter and the 6/5 Shapeshifter-Golem. There are now four Unstable Shapeshifters, two copying 3/3 Golems and two copying Inverters.
Now we can begin the full loop:
- Cast a Saw pair on a Shapeshifter-Inverter and the original Golem.
- Cast Boulderbranch Golem. (net -1 life) All Shapeshifters trigger.
- After each of those Shapeshifter triggers resolve, cast a Saw pair on an Inverter and the 6/5 Shapeshifter-Golem. The first two times, it has to target a plain Inverter; each subsequent time can target a Shapeshifter-Inverter that does not have a waiting trigger to become a 6/5 Golem.
The net result is -1 life, and +1 and then *3-2 to the number of Shapeshifters, which works out to *3+1.This can be repeated 7 times, ending with 7 life and 9841 Shapeshifters.
Cast the final card, Life of the Party, from hand. All the Shapeshifters turn into that.
Attack with all possible creatures, getting triggers from Life of the Party and the Shapeshifter copies of it; put the original's trigger on the bottom.
Cast a Saw pair on an Inverter and the original Golem, this time letting the triggers turning Shapeshifters into Inverters resolve first, so that afterwards they become Golems and get to stay Golems for a while.
A trigger from Life of the Party resolves, giving a big boost to the power of a Shapeshifter that is now copying a Golem. Start Sawing again with that one.
(As before, leave the high-value Shapeshifter triggers on the stack until they are ready to be consumed. Now that there is excess life being gained from the big Golems, it can be spent on extra Saws on small Golems. This should work out to a number of Knuth arrows somewhere around the base-2 logarithm of the starting power.)
Repeat with each Life of the Party trigger, until the final one resolves for the original Life of the Party, which is left to deal the big damage.
Some other things I found:
Also attaining F_{w+1}, but not as high, and only on the draw: Black Lotus, Channel, Chromatic Orrery, Unstable Shapeshifter, Mycoid Shepherd, Saw in Half, Soulfire Grand Master, Life of the Party
Some cards with dual-purpose potential: Conclave Mentor, Packsong Pup, Swiftgear Drake
There are also a few cards that can remove a spell from the stack and put it back on, which is similar to moving it to the top: Narset's Reversal, Psychic Rebuttal, and Guile.
I didn't think the extra cards could be assumed to be Wastes?
(emphasis added)
Another benefit of Aluren is that it costs only 4 mana, but I couldn't find any way to get a big advantage from that; the best I have is starting with Black Lotus and Sunscorched Desert for 1 extra damage. (Crystal Dragon looked promising, but there's no way to get the required mana to cast it from exile without giving up multiple Goblin tokens.)
For 3 cards, Parallel Lives, Astral Dragon, Mirage Phalanx is still the best I know of.
For 2 cards, Devilish Valet and Displacer Beast is better. All three default dungeons have unavoidable limiting effects: "Each player loses 1 life" in Tomb of Annihilation, and card draw at the end of Lost Mine of Phandelver and Dungeon of the Mad Mage.
Start by casting Displacer Beast four times to venture through Lost Mine of Phandelver, getting a Goblin token at Goblin Lair, proceeding from there into Storeroom, and drawing Devilish Valet at the end. Cast Devilish Valet. Then go through Tomb of Annihilation 19 times. At Oubliette, sacrifice the Goblin token the first time, and sacrifice The Atropal on later times. This triggers Devilish Valet 19*4=76 times. Finally, venture six steps into Dungeon of the Mad Mage, going through Muiral's Graveyard, for 8 more triggers of DV, making its power 2^84, and attack.
4 cards: Black Lotus, Channel, Skitterbeam Battalion, Tribal Unity with X=8 attains 36 damage.
5 cards: Black Lotus, Channel, Dryad's Revival retrieving Black Lotus, use it for green mana, flashback Dryad's Revival retrieving Black Lotus, use it for red mana, Nylea's Colossus, Rionya, Fire Dancer. Go to combat and make 4 hasty copies of Nylea's Colossus, producing 20 of its triggers, all targeting one of the copies. Attack for 6*2^20+18=6291474 damage.
- One permanent can be sacrificed to bounce Kaya, Geist Hunter and double the current token multiplier.
- One Whale produces many Krakens, for an exponentiation.
- One Fish produces many Whales, for a tetration.
- One Reef Worm produces many Fishes, for a pentation.
- Exiling Reef Worm to Dollhouse of Horrors produces many of them, for a hexation.
When the original Reef Worm is sacrificed, it produces two Fishes; after those are fully consumed, the token multiplier is >2^^^4. Then Dollhouse of Horrors makes >2^^^4 copies of Reef Worm, for an end result of >2^^^^(2^^^4). Hold on to one of the Reef Worm tokens, and after the last Whale is sacrificed, start using the first ability of Kaya, Geist Hunter to put +1/+1 counters on it, so that it can attack for >2^^^^(2^^^4) damage.Unfortunately, this can't be made into a 7-card solution by adding Black Lotus, Channel, and Lich's Mirror, because the token-doubling effects persist through Lich's Mirror's effect.
- 2 PG triggers. First copies LF 5 times.
- 1st LF copy gives 6 life and one Golem is destroyed. Activate CA targeting PG.
- 2nd LF copy gives 8 life and one Golem is destroyed. Activate CA targeting PG. 1 mana left over.
- 3rd LF copy gives 10 life and one Golem is destroyed. Activate CA targeting PG. 4 mana left over.
- 4th LF copy gives 12 life and one Golem is destroyed. Activate CA targeting PG twice. 2 mana left over.
- 5th LF copy gives 17 life and one Golem is destroyed. Activate CA targeting PG twice. 5 mana left over.
- Second PG trigger copies LF 21 times.
- etc.
For the final card, like before, we want an instant/sorcery that can target a Golem and draw a card (or otherwise obtain Lantern Flare from the library to be recast), but that's not all this time. The Golems are being destroyed because of Lantern Flare's damage, which normally makes the final spell fizzle. We want to choose a card that can avoid fizzling.Black Lotus, Channel, Precursor Golem, Cogwork Assembler, activate once targeting Precursor Golem, Academy Manufactor, Confront the Unknown... but that goes infinite by using Cogwork Assembler to copy Clues/Foods/Treasures.
Adrix and Nev, Twincasters, Ratadrabik of Urborg, Cabaretti Confluence... but the haste isn't copiable. If it were, we could make 2 copies of A&N, make 8 copies of A&N, and then make 2048 copies of Ratadrabik, and get 20490 Ratadrabik triggers on A&N and a bunch on Ratadrabik, and deal >2^^20492 damage.
Instead, Kindred Charge is the best I could find, dealing >2^^8 damage.