Minecraft has the redstone wires mechanism that can be used to build circuits. Is Minecraft Turing-Complete, i.e. can it be used to simulate a Turing Machine (if we ignore the problem of infinite memory)?
Notch himself has said in an interview that yes, the Redstone blocks in Minecraft allow construction of Turing-complete Machines.
A couple people have even constructed ALUs and CPUs, for instance the following one. The creator was planning on adding a memory array to allow programming it.
I'm afraid that any finite-sized redstone building (even in an infinite world) can only store as much bits of data as the amount of redstone put in it, therefore it's not Turing Complete.
If you're talking about infinite-sized redstone buildings, well, you can quite easily build conway's game of life in minecraft, which is turing complete. The "quite easily" won't work if we were in a 2D Minecraft space, and there, well, that's an interesting question :)
Here's a neat example of an implementation: