The dissertation project of theoretical mathematician Professor Dies Yaws Githe, recovered from the depths of the MEGA-MEGA-YETI-HAND section of the Infinity Hotel. The die appears as an iridescent d20 of some kind of bone or ivory, of the typical size and dimensions you'd have available at the table. The grain of the bone on each of the sides appears to shift and warp under non-rolled conditions.
Professor Dies theorized it was made from the long tooth of the fabled Recursive Infinicorn. Under normal circumstances, when the die is rolled it will produce a symbol on its topmost face which in all likelihood represents some impossibly-too-large-to-understand number. Staring at the number for even a short time produces headaches and nausea in little tiny apelike human brains. It is infinitely impossible that the same symbol will ever appear again.
Under unusual probabilistic circumstances when something is fiddling with fate, such as the influence of a god, a probabilistic entity, or a mathematical determinator, something else is likely to happen entirely. As such, the Professor Dies discovered that the die can be used much like a canary in the coalmine for the proximity of anomalous probabilistic influences.
The ∞ Die Has Been Cast! (d20)
(And circumstances are NOT normal!)
1. The Die keeps rolling and rolling, faster and faster, heating up itself and the air around it. If not restrained it will bore through the earth and produce a thermal explosion about 1 to 20 meters below the surface, producing the effects of an Earthquake spell. It can be painstakingly recovered. A restrained infinitely-rolling die can probably be made into an everlasting reactor, or something.
2. The Die splits into two dice, each of exactly half the volume and mass as the original. Every 1d20 seconds it will do this again. Unless time is somehow reversed it will continue to split until it becomes a pile of Recursive Sand the same mass and volume as the die. Anything that eats, breaths, or looks at this sand for a modest amount of time effectively becomes a living Infinity Die, and will function as the die would in similar circumstances. It is up to you as to what 'being rolled' encompasses. If the Recursive Sand is scattered into an environment it will have similar results upon the environment, resulting in weird probabilistic nonsense like trees growing into squares or water flowing against gravity.
3. Higher dimensional projections fail and the Die puffs out becoming a sphere of equal mass and volume. Hypothetically there is an infinitely small die face oriented up at any given time, but in all likelihood it's going to start rolling downhill.
4. The Die splits into two dice, along with the one who rolled it, producing a Doppelganger of the roller who in any given situation will do the opposite of what the player-character does, even if it's obviously self-destructive. All future results of each of the dice have half the normal number of outcomes (d4):
- Evens + Odds. Easy enough. Effects this Table's results.
- One to Halfway + Halfway to Infinity. Virtually functionally identical.
- Infinity + Infinity x 2. Second die rolls twice on this table.
- All Digits Ending in 1-5 + All Digits Ending in 6-0. Effects this Table's results.
If this result and its sub-table eventually produce an impossible combination, then a Paradox Archangle (sic) will come and beat the shit out of you for messing with reality.
5. For the next 5 rolls in which it's applicable: X in Y chances are now Y - X in Y chances. (Ex: 1 in 6 becomes 5 in 6.)
6. The digits shown seem like a summoning circle. Whatever is messing with your fate spontaneously appears out from the die in reverse-spaghettification. If it's a god then they will be present in the flesh, and they might need to explain themselves, or you could possibly murder them forever. If it's a probabilistic entity it'll probably be pretty confused as to why it just got reverse-sphaghettified, and possibly meander a bit or eat the luck of everyone present. If it's a probabilistic determinator, then congratulations! You now have a new probabilistic determinator in your possession! If none of the above seem plausible, then The Devil appears, laughs, and offers to buy back his die for deferred sentence in Hell. Refuse his third offer and he'll simply take it.
7. Strange number! For the next 7 dice rolls not related to the Infinity Die, the roller can determine their results.
8. The next time the roller rolls a standard die roll for your system, they roll two dice and add them together to generate the result.
9. The Die fumbles and rolls out of existence. To the lay observer it will have simply disappeared. It's on some higher plane now! Can be retrieved by Astral Projection or similar plane-shifting.
10. The die becomes as a mirror face, and a shady individual appears within it. They ask for someone to kill, and require only a name. Given a name, the Mirror Assassin, who moves in reflections, will go and kill this person provided there are any reflective surfaces for him to use. He is very patient, capable of waiting centuries for a time to strike.
11. Professor Dies Yaws Githe spontaneously appears! Alongside him is a rotting worm-infested corpse that looks like it just came out of a shallow grave (because it did). Professor Dies will be momentarily very confused, as he will have just been teleported from whatever he was previously doing. 1 in 6 chance he's indecent. After coming to his senses, he'll realize what happened: "You rolled EPSILON-DEVILFACE-DEVILFACE again. Well, welcome to the club you poor sod!" This die roll summons anybody who ever rolled this number, dead or alive. Professor Dies unfortunately had to put down this poor nameless bloke, who believed he was an evil ghost and tried to kill him. It was self-defense, he assures you!
If circumstances are amenable, Professor Dies can explain the Infinity Die and its strange mathematics. Unfortunately he can only do so in terms that someone with at least a graduate degree in fantasy theoretical mathematics could understand, so it's best to gloss over it 'off camera'. He can also explain that he is currently researching why he isn't being constantly summoned forever: if infinity exists, then infinite instances of this die roll being rolled should exist, and he (and you) should be being teleported all over infinity forever. This apparently isn't the case. He best hypothesis involves ghosts.
12. The Die result is 12. The odds of this happening naturally are 1/∞. Incredible! The next 1d12 die results on any die that can roll a 12 rolls will be 12, including on the Infinity Die and the d12 rolled to determine the number of twelves. The odds of this happening naturally are considerably less than 1/∞, which really calls into question the principles of probability in general. If somebody comments that that initial 12 really shouldn't count, since it means there are only really 11 rolls that become 12, then you are obligated to tell them they are stupid and unnecessarily argumentative!
13. The number rolled gains the attention of a Super-Planar Devourer, whose projection appears in the form of a mass of slavering mouths which will latch onto and eat anything they touch, sending the bits and pieces bitten off scattering through higher dimensions to be lost forever. This can destroy literally anything in this reality. If the roller didn't cast the die away from them for the initial roll they must Save or lose a limb. The Devourer will linger in place for 1d6 Days before unlatching from reality. For the Devourer, this is the equivalent of licking a mossy rock for sustenance. Great for destroying undestroyable things!
14. The roller gains a glimpse of the future on the die face. At the roller's choosing, they to retroactively act upon an event it as if they had known about it one minute prior to it happening. What they did cannot prevent the event from happening, but it can give them some sort of advantage.
15. The Grim Reaper appears, and informs you that you have just wagered your life against your death. The roller chooses a game of chance of their liking. If there's a house advantage, then the Grim Reaper gets to be the House. Win the game, and the roller is functionally immortal (though not eternally young or abled, see: Tithonus). Lose, and the Reaper instantly claims his due.
16. The roller gains a glimpse of a strange number and gains strange insight into the nature of mathematics. They can now comprehend very very very very very very large numbers, and the roller doesn't get sick when looking at them, such as under 'normal' rolling circumstances for the Infinity Die. Additionally, they now comprehend and remember all arcane symbols and can ascertain their meanings, and they are no longer affected by the effects of magical runes unless they want to be.
17. The Die result is 17. The odds of this happening naturally are 1/∞. Incredible!
18. Chosen roll! Every subsequent time the roller naturally rolls a '18', they bank a roll of 18 which can be used in place of any future roll. You can only have one '18' banked at a time, unless you roll this result again on the Infinity Die without using a banked 18, in which case the number of banked '18's goes up by one.
19. God appears (or whatever passes as closest to God in your setting) and commands that you roll the die again. Technically speaking, you have free will, so you could refuse, but that probably wouldn't go well. I suppose if you have something to say to God now is the time. If you refuse three times then a Saint will appear and snatch up the die, using the good fortune of God's blessings to escape.
20. Wish Number. Gain one Wish.