A nice puzzle which I found in this Math Overflow page: Is there a position with a finite number of chess pieces on an infinite chessboard, such that White has a forced win in ω moves? The meaning of this is that White has a move such that, for every possible response of Black, White has a guaranteed checkmate in a number of moves bounded by a finite number N; but before Black's first move, we cannot put a bound on how large N might be.

The thread gives a solution, and also links to this paper, where higher ordinals and questions of computability in infinite chess are also considered.