Some chess variants
3d chess (with 8 boards) but each board is rotated 90 degrees vs the one above it (so traversing z twists you into a different context). Possible moves have forward replaced with down. All boards are fully loaded and turns rotate round robin between boards (each makes a move on first board, then second, etc). Pawns that make it to the far side of the lowest board become queens. Winner is the first to take (not check) 5 kings.
You have 8 boards. (Each board is adjacent along one of 8 dimensions.) Each move involves a move on each of the 8 parallel games, but a valid move is to swap a piece with another (adjacent) board. Swapping occurs by switching with a piece on the same spot on another board (or, if there is no piece there, swapping with the notional ‘empty space’ piece).
Winner is first to take (not check) 5 kings.
(The kings are taken because checkmate is hard to calculate when any piece can spontaneously move into hyperspace.)
If you want to make it more interesting, you can have black move first on even-numbered boards.
(“8–2” chess would have a second order of 8 boards adjacent to each of the first 8, for 64 boards. Still playable but harder.)