The CPU loadis going to depend heavily on a player's projected maximum range, as well as the complexity of the operations that each tile performs. If it's just a question of finite versus infinite, you could just postpone that decision until you have some rough version of what each square does each turn, then make a program that measures the time taken by a calculation on 1k, 1M, 1bln etc. squares (that's cubes 10, 100, 1000 on a side*); you'll be able to find your feasible range that way, and determine for yourself whether that's something that you expect your player to reach. (also, note that it's probably fine to use a pretty streamlined version of AI for far-away squares, and possibly "frame-skip" on them or draw them every five turns or what-have-you).
* note that this doesn't mean you'll find a measure of the maximum side length, since it's, erhm, [i]unlikely[/i] that your player will hit a billion squares in a single game, ever.