Checkers (draughts) uses only half the squares on the board, begging some way to make the other half useful. Vladimir Vigman proposes two independent games. Similar ideas:
Two independent games of different sizes, one 10x10 and one 8x8, to prevent mirror strategy.
Two independent games played by two pairs of opponents (4 people total) cross ways. The appeal/difficulty is dealing with the distraction of another game going on on the same board. Fast time controls would add a physical challenge of needing to avoid hands colliding.
Something like bughouse chess where captures on one half-board can be transfered to the other half-board.
A way for online spectators to watch two independent games simultaneously making efficient use of screen area. Online could easily also provide it as a way to simultaneously play two different opponents.