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Of course, there are some variants on these pages played on infinite
boards.  I have thought a bit about infinite-dimensional games as well.

I suppose any large size is workable, especially if pieces have short
moves: the game becomes a large map, which you could zoom in on a sector of
and make relatively normal-looking moves.  (It becomes a discrete version
of many RTS games.)  Infinite-dimensional doesn't admit as good a visual
tool.  :P