Bughouse can be played as a two-player game (instead of four-player), with each player overseeing two boards. Also, Bughouse could be played with more than 2 boards: using the standard set up, captured pieces can only be dropped on boards in which you play the captured color. With pieces that can flip color (perhaps tiles printed with opposite colors on each side), captured pieces can be dropped onto any board.
Playing many boards simultaneously likely captures the feeling of being a field marshal overseeing many battlefields. You are responsible not only for making the right move on within each board, but committing the right resources (captured pieces) among boards.
A set up with the boards stacked in a column is better suited for play with two players. Then the game begins to resemble 3D chess, with pieces traveling between boards.
For even more dimensions, consider multiple cubes.