Versions 1-4 of my Turing machine were built around Wolfram's (2,3) machine as proved universal by Alex Smith. None of them provided the infinitely nonrepeating tape required for true universality of this machine, so they all actually fail to be proper universal Turing machines. At best they can be considered proofs of concept or works in progress. They would function as Turing machines, but not universal ones.
They all used Teysa, Orzhov Scion and Mind Bend to trigger off tokens of three different colours dying. Magic only has five colours, so when creating version 5 I was moving from a machine with three colours to a machine with 18 colours, I had to stop using Magic colours and start using creature types instead.
Version 1 had the basic core of Kazuul Warlord, Noxious Ghoul and Time and Tide. But it used Isochron Scepter to cast Time and Tide. It arranged to untap the Scepter at a particular point and required the player to choose to activate it whenever possible. I was sure I could do better.
Version 2 of the Turing machine used Spellbinder to cast Time and Tide in a triggered way. It had a very complicated setup using Grand Melee, Hellkite Charger, Vulshok Battlemaster, No Mercy, Second Sunrise and two Spellbinders, amongst other cards (!), to have a combat step, force certain creatures to attack, and then resume processing afterwards. The details are on this Draw3Cards answer.
Version 3 of the Turing machine used Toshiro Umezawa to cast Time and Tide. It used Riftsweeper to bring the exiled instant back into the graveyard where Toshi wants it, and Skirk Drill Sergeant and Wheel of Sun and Moon to repeatedly summon the Riftsweeper.
Version 4 of the Turing machine used Chancellor of the Spires to cast Time and Tide. It was the version on this site when it caught the internet's attention in September 2012. It used Skirk Drill Sergeant and Wheel of Sun and Moon to recycle the Chancellor, which meant it needed Carnival of Souls and False Dawn to allow the Skirk's mana payment to happen.
So what might version 6 hold?
The most inelegant part of the Turing machine version 5 is the way that Alex is repeatedly given "may" options that he has to say "yes" to. It'd be much better if we could arrange that no player receives any choices, no player has any option at all.
This would allow us to prove, for example, that there are some Magic games which might be stuck in an infinite loop (the machine never halts and the game is a draw), or may terminate (in a victory for one player), but which is the case depends on, say, whether there are any even numbers not expressible as the sum of two prime numbers, or some other unsolved problem in mathematics. As it is, in versions 1-5 player A always has the choice to start refusing to put counters on when the Kazuul Warlord offers him the choice, or decline to change state when Chancellor of the Spires gives him the chance.
So is it possible to assemble a Magic Turing machine where no player ever gets offered a "may" option? I don't yet know. I'm working on a couple of possible answers. It's quite a bit harder, but it's not clearly impossible. Cathars' Crusade could replace Kazuul Warlord, and Omen Machine or Wild Evocation could replace Chancellor of the Spires to force Time and Tide to be cast, but they have their own complications. Some of my progress is being discussed on this thread on the Wizards of the Coast forums.
If you'd like to be notified when I find a way to do this, or make other updates to the Magic Turing machine site – or if you have suggestions! – then email me!