[tromp]

That probably involves creating the maximum number of pieces, by capturing 4 pawns to promote the 12 others. Let's give each side 1 king, 3 queens, 3 rooks, 3 knights, 2 light squared bishops, and 2 dark squared bishops.

Then have these move through as many board permutations as possible without triple repetition. This would give a multinomial of (64 choose 36 1 3 3 3 2 2 1 3 3 3 2 2) ~ 4.6 x 10^41 positions to move through, which could be multiplied by 2 for side to move (over estimating a lot since many positions will have the wrong king in check, or the right king in impossible check) and another factor 2 for first repetition, giving roughly a 2e42 upper bound.