Consider playing go 囲碁 on multiple disconnected quarter-infinite boards. 4 boards vaguely mimics the traditional game on a square.
Vague idea.
The object of the game needs to be modified because one will never run out of space to keep playing more stones. Perhaps the goal is to capture a majority of corners (need an odd number of boards). Or create one or more immortal stones or groups, groups which cannot be captured by the opponent even if you pass forever (probably 2 eyes).
Infinite ladders are a problem.
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