The largest I could find was for Double Chess, with a 16x12 board. Is there a variant with a bigger board? It doesn't have to be a variant in which chess pieces follow their orthodox movement rules.
Taikyoku shogi should be a good candidate. It was created in the 16th century, so it's a variant that has actually been played. The board size is 36x36, and it has 209 different types of pieces.
From wikipedia (emphasis is mine):
Infinite chess is any of several variations of the game chess played on an unbounded chessboard. Versions of infinite chess have been introduced independently by multiple players, chess theorists, and mathematicians, both as a playable game and as a model for theoretical study. It has been found that even though the board is unbounded, there are ways in which a player can win the game in a finite number of moves.
Another contender is Kubikschach. A 3-dimensional variant played on an 8x8x8 board (512 squares total)
Suppose we only allow chess pieces (so excluding all shogis) and only a 2-dimensional board. Then there are:
- Double Chess (II) (Hayward, 1916) (16 files x 12 ranks = 192 squares)
- Fortress chess (16x16 minus 8x2 off the centre of each side = 192 squares)
- Four-handed chess (many variants use a board 14x14 minus 3x3 off each corner = 160 squares)
s: David Pritchard. The Encyclopedia of Chess Variants. pub. Games and Puzzles Publications.