EDIT: with the Companion errata, this no longer works.
For clarity, this is just a cheerful thought experiment.
In order for this deck to work, we break two minor rules. Shahrazad is now legal, my orders. Second, either your commander doesn't have to follow Companion deck-building rules, or there's a magical imaginary Legendary Creature with five-colour identity and a clause saying that Companion deck-building restrictions don't apply to it. I consider the second the most likely.
You also need to know that in each sub-game, you can declare a new Companion.
So here's how the deck works:
There are hella piles of tutors to fetch anything and everything you need. There's also quite some redundancy.
Create infinite mana, probably with Devoted Druid and Vizier of Remedies. It's helped with Purphoros, Bronze-Blooded.
Get a Leyline of Anticipation on the battlefield, so you can do everything at instant speed. Alchemist's Refuge also works.
Cast Shahrazad, and then infinitely copy it with Nivix Guildmage.
Use Riftsweeper and Pull from Eternity to pull everything out of exile, with Archaeomancer and/or Capsize to recast them.
Elixir of Immortality shuffles everything from your graveyard into your library.
Azure Mage can draw your whole deck.
Wheel of Sun and Moon, enchanting yourself, along with a discard outlet such as Fae of Wishes, can get a precise and specific set of cards in your library.
Now here's where the fun starts. For every companion that is EDH viable - Gyruda, Doom of Depths, Jegantha, the Wellspring, Kaheera, the Orphanguard, Keruga, the Macrosage, Lurrus of the Dream-Den, Obosh, the Preypiercer, Umori, the Collector, Zirda, the Dawnwaker - there is a pile that wins you the subgame on turn one. This is presuming no interaction from your opponents, but there's only so much magic I can pull off.
Each pile is separated into custom categories, and I suggest you try and work out how each wins. Each is shown beneath individual spoilers, good luck.
Gyruda, Doom of Depths.
The only tricky part to this pile was making it deterministic.
Start the game, and put Leyline of the Void into play.
Play Plains and Silence. This isn't necessary, but there's no reason not to.
Make six mana with Lion's Eye Diamond, Mana Crypt and Mox Opal. Lion's Eye Diamond will make you discard Gigantoplasm and Kozilek, Butcher of Truth, but they'll be shuffled back into your library with Kozilek's trigger, along with Lion's Eye Diamond and Silence.
Cast Gyruda, milling Gigantoplasm and Kozilek, Butcher of Truth. Lion's Eye Diamond will also be milled, as will Silence, but you can ignore the latter two, they are no longer relevant. Gigantoplasm comes in off Gyruda, copying Gyruda. State-based actions ensure that you have to bin Gyruda, and then two more triggers go on the stack. Kozilek's shuffle trigger and Gyruda-clone trigger. Put the Gyruda trigger on the stack, then Kozilek. Resolve Kozilek, shuffling in Gyruda, Kozilek and Lion's Eye Diamond. Resolve Gyruda-clone trigger, milling Kozilek, Gyruda, and Lion's Eye Diamond. Repeat.
You get infinite mill, with Leyline of the Void sending opponent's cards to exile to ensure no Eldrazi triggers of their own.
Kaheera, the Orphanguard.
This one was extremely hard. It also, unfortunately, doesn't rely upon Kaheera, but it turns out just a vigilance lord is hard to combo around. Kaheera is still a legal Companion for this pile, though if anyone can do better I'd love to see it.
Mountain, Mana Crypt, Seething Song and Geosurge. Eight red floating, of which seven must be spent on creatures. I'll represent this "creature mana" as R(cr).
Brighthearth Banneret, Runaway Steam-Kin, RRRR(cr) R floating.
Cast Grinning Ignus, RR(cr) R floating, one counter on Steam-Kin.
Activate Ignus, using the unrestricted red mana. Add RCC to mana pool. RR(cr) R CC floating.
Cast Grinning Ignus, using R(cr) C, two counters on Steam-Kin. R(cr) R C floating. Again, activate Ignus, then re-cast. Three counters on Steam-Kin, R CC floating, all now unrestricted. Remove the counters from Steam-Kin to add three red, then continue casting and bouncing Ignus. You will slowly net infinite colourless mana.
You can then sink this into any card really, but I chose Goblin Cannon because it's truly an awful card.
Keruga, the Macrosage.
Oof. What a restriction. Leyline of Sanctity is not a necessary card, just minor protection as you end up going to 1 life.
Reveal Chancellor of the Tangle, and make the rest of your mana with Crystal Vein, Elvish Spirit Guide, and Simian Spirit Guide. GG R CC floating. Play Treasonous Ogre, leaving G floating. Pour your life into the ogre, to cast Gemstone Array, filter it, and cast Laboratory Maniac then Keruga, the Macrosage.
Lurrus of the Dream-Den.
I love this card. Love, love, love it. Will probably build it it's own deck.
Play Ancient Tomb, Lion's Eye Diamond, Lotus Petal, Mox Opal. Tap Opal for mana before saccing Petal. CC BB floating, pre-LED.
Play Carnival of Souls, CB floating. Play Carrion Feeder, triggering Carnival, adding B back again. Play Zulaport Cutthroat, triggering Carnival, B floating.
Crack LED, throwing Kaya's Ghostform into the bin, and then play Lurrus of the Dream-Den. Lurrus triggers Carnival, BB floating. Use Lurrus' ability to cast Ghostform on Lurrus. Sac it to Carrion Feeder, triggering Zulaport Cutthroat. Ghostform will return it, and as a new Lurrus, you can re-target Ghostform. Carnival will trigger and give you the mana necessary to continue looping, whilst Cutthroat keeps you life-neutral and kills off your opponents.
Obosh, the Preypiercer.
I struggled with this one, mostly for the mana required. Heartless Hidetsugu was the obvious win-con, though I considered Blasphemous Act and Boros Reckoner, but unfortunately Reckoner targets.
So. The actual combo. Reveal Chancellor of the Tangle for G.
Play Fiery Islet, tapping for U, and putting our life total to 39. This is important.
Cast Allosaurus Rider off Chancellor and Elvish Spirit Guide. Play Energy Tap on Rider, for seven colourless. Use the G off Chancellor, cast Metamorphosis, saccing the Rider. Add eight R to your pool.
Seven C, eight R floating. Cast Heartless Hidetsugu, Obosh, the Preypiercer and a haste granter. I chose Purphoros, Bronze-Blooded. As long as it's a creature, and costs five mana or less, and requires three or less red mana, you're gucci.
Activate Hidetsugu, and everyone except you dies, as we're on 39 life.
Umori, the Collector.
I wasn't looking forward to this one. Sorceries, for Storm? Creatures for a weird combo? The difficulty with both of those is mana availability. Then I realised artifacts would be absurdly easy. Turns out, it was.
Play and sac Crystal Vein, then play Lion's Eye Diamond, Mana Crypt, Mana Vault, Myr Retriever and Ashnod's Altar. One mana left. Crack Lion's Eye Diamond, discarding Walking Ballista and Junk Diver.
Play Umori, the Collector, choosing artifacts (obviously). Sac Myr Retriever, getting back Junk Diver. Cast Junk Diver, then sac to get back Myr Retriever. Repeat, generating one colourless mana each time. Once you have enough, sac whichever creature you currently have to get back Walking Ballista, then shoot everyone to death. Again, any infinite colourless mana sink will do here, as long as it's an artifact. If you're pushed for main-deck slots, you could even use Goblin Cannon again here.
Zirda, the Dawnwaker.
Oh, this card is laughably broken. I think I spent less than a minute brewing this pile, and of all of them, this one should be easiest to work out.
Lotus Petal, Mana Crypt, Mox Opal, and Simian Spirit Guide. RRR CC floating. Play Basalt Monolith, tap for the three colourless, then play Zirda, the Dawnwaker. With CC floating remaining, you can untap and tap Basalt Monolith for infinite colourless mana. Sink it into Walking Ballista, and bam. Easy.
Worth noting that this pile is only six cards. Maybe throw in a Plains and Silence? I leave that up to you.
