A video of the solution of a Rubik's cube of side 100, from YouTube.
I don't know the details of the algorithm, but it has the interesting property that at first it looks like nothing's happening, and then larger and larger blocks of each color seem to form -- I'm not sure if this is because of the low resolution or if this really is some sort of "phase transition" in the algorithm. This is the sort of thing one just doesn't see in the normal-sized cube.
I came to this from a solution of the size-20 cube, which works in a much different way; each of the faces of the cube gets "filled in" in turn. So in the second case the algorithm appears to be making steady progress; in the first case it doesn't. There are essentially two different families of algorithm.
It would be interesting to know how the solution time (measured in number of moves) of these families of algorithms -- and I'm calling them "families" because I suspect that the actual side length of the cube isn't all that important -- scales with time. Also, how do these algorithms compare with the asymptotically fastest one? One could compute a lower bound on the number of moves needed to solve a Rubik's cube of arbitrary size by just computing the number of possible positions (analogously to the calculation of that number for the normal cube I outlined here) and comparing that to the number of possible moves one can make from a given position. An absolutely useless question -- is that lower bound (I worked it out once, but I forget what it is, and it was the sort of the back-of-the-envelope work that might have been wrong anyway) asymptotically tight? That is, does there exist a solution procedure for the Rubik's cube for which the number of moves needed to solve the size-n cube is within a constant factor of this obvious lower bound?
As for what that lower bound is, I may write that up as another post; I'd either need to find the notes I made on it a few months ago or (more likely) rederive them.