Played on an unbounded chessboard, “infinite chess” feels familiar but can lead to some strange situations. In this position White threatens to give checkmate by advancing his rook one square. Black can avoid this fate by throwing his rook any finite distance to the north, as shown. Now when White’s rook gives check, Black’s king has room to retreat up the board. But it’s still curiously doomed: By advancing alternately his queen and then his rook by two squares each, repeatedly, White can now harry the Black king indefinitely northward after the departed rook. Because each of White’s moves gives check, Black’s king has no time to do anything but flee, and when it fetches up against the far-flung rook White will finally give checkmate.

Thus, in the diagramed position with Black to move, it’s accurate to say that White has a forced checkmate but that there’s no bound on the number of moves it may take to deliver it.

(C.D.A. Evans and Joel David Hamkins, “Transfinite Game Values in Infinite Chess,” arXiv preprint arXiv:1302.4377 [2013].)