The upper bound is a few thousand. A game is considered drawn if no pawn has moved and no piece has been captured for 50 moves. And there's also the threefold repetition rule: if the same exact position (counting things like castling eligibility etc) occurs three times it's drawn.
I think the upper bound is 6300, then. Each pawn can move 6 times, times 16 pawns, times 50 moves before a game ending draw, plus each capturable piece that can delay the game another 50 moves (15 on each side)
