[AlexTFish]

On the "Cards" page, http://www.toothycat.net/~hologram/Turing/Cards.html:

> The first sixteen Reanimators were cast in the normal fashion, four by each of the players, and donated by a Bazaar Trader. The other twenty are Clones and Vesuvan Shapeshifters copying a Rotlung Reanimator. The phasing enchantments can include Teferi's Curse, Cloak of Invisibility, and Copy Enchantment, cast by any player and donated by a Bazaar Trader. These creatures and Auras must not be tokens, because tokens disappear when phased out.

[k__]

So to be Turing complete you have to play with 4 players?

[AlexTFish]

The machine as demonstrated on that page requires 4 players. There are probably ways to do it with fewer, but you might have to accept situations where two abilities trigger and the players need to choose an order to put them on the stack (though it makes no difference either way).