Based on comments from the 2048 thread, I'm running a deterministic variant of this which just cycles Left, Down, Right, Down.

After an hour or so, the largest tile is 524288, 2^19: http://i.imgur.com/Yk758sx.png

While I was writing this comment, it collapsed down to the next largest tile, 1048576, 2^20: http://i.imgur.com/PVPCvUC.png