A 2x2x16 circular chessboard has only 64 squares, and of course it is "impossible" to have a decent game of three dimensional chess in only 64 squares.
The reason for this supposed impossibility is that at least one dimension of the board must be so short that diagonal movement will be too limited; but the "impossibility" can be overcome by using Crooked Bishop instead of Bishop and Crooked Queen instead of Queen.
The following UAD shows the opening position of 2x2x16 Race Chess. Note that part of the board is repeated because doing so makes moves easier to visualize. The White King is on the square 16b1 and the Black King is on 8b1.
2a . * P R R P . * . * p r r p . * . * P R R P . * 2b * . P N N P * . * . p n n p * . * . P N N P * . 13 15 1 2 3 4 5 6 7 8 9 11 13 15 1 2 3 4 14 16 10 12 14 16 1a * . P B B P * . * . p b b p * . * . P B B P * . 1b . * P K Q P . * . * p k q p . * . * P K Q P . * 13 15 1 2 3 4 5 6 7 8 9 11 13 15 1 2 3 4 14 16 10 12 14 16
For example, on an otherwise empty board White's Crooked Bishop could go from 1a1 to 1b2-1a3-1b4 and so on (but is forbidden to return to its square of departure and thus make a null move).
Alternatively, it could move 1a1-2a2-1a3-2a4 and so on, always on squares of its original color.
A Corkscrew Bishop would move widdershins from 1a1-1b2-2b3-2a4-1a5 and so on or would move clockwise from 1a1-2a2-2b3-1b4 et cetera. Beginners should probably avoid using this piece, but players who are accustomed to playing on the 3D board should enjoy 2x2x16 Corkscrew Race Chess.
Knights are weak in this game. Fibnifs might be a better choice. There has as yet been no playtest, so who can say?
Monochromatic Alice 2x2x16 Race Chess would be a game in four dimensions, and therefore could not be played by mortals.
The small board also means that squares cannot be defended by more than two Pawns at once. This is good! On larger 3D boards, Pawn masses can be too difficult to break up.
Thanks to the small board, the game can be played in a reasonable number of moves; on the 8x8x8 board, an average game should probably go on for more than 300 moves.
Two additional advantages of the small board are that the 3D play will be easier for 2D players to visualize, and that making a physical 2x2x16 circular board is much easier than making an 8x8x8 board.