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.
Let's assume the 50-move rule is automatically applied. We can have 49 moves per pawn move, followed by 49 moves per piece captured. A maximum of 8 pawns (4 white and 4 black, for example) can possibly be advanced over the course of the game to the opposite side (6 squares ahead from their home square). Therefore, it seems to me that the maximum number of pawn moves could involve one player advancing their pawns 4 spaces, followed by the opposing player capturing all those pawns (using pieces rather than pawns, as both moving a pawn and capturing a pawn would be a waste), and then advancing their pawns to the back rank. This would be 8x4 + 86 = 80 pawn moves, plus 8 pawn captures. That's 88 allowances for 49-move knight sequences, plus the initial 49-move knight sequence made from move one. So, with no pawns remaining, and all of black's pawns converted to pieces on the back rank, we have 8949=4361 moves. Now, there are 22 pieces and 2 kings remaining on the board. That puts an upper limit of 22 captures, leading to an additional 2249 allowable moves. All together, that is 11149=5439 total moves. I'm probably off by a few.
> A maximum of 8 pawns (4 white and 4 black, for example) can possibly be advanced over the course of the game to the opposite side (6 squares ahead from their home square).
Not so! Pawns move diagonally via capturing, which theoretically could allow white to promote all 8 pawns just in the d column, and black to promote all 8 in the e column (at the expense of various other pieces).
edit: of course, you'd run out of pieces first, but you can accomplish something similar by using multiple columns.
Sample game start that accomplishes a subset of this (2 pawns promoted from each side, 6 pawns remaining from each side):
> A maximum of 8 pawns (4 white and 4 black, for example) can possibly be advanced over the course of the game to the opposite side
I think they all can.
If white’s A pawn takes a black piece in the B file and black’s B pawn takes a white piece in the A file, both white’s two B pawns and black’s two A pawns have a clear route to promotion.
You can do the same for files C and D, E and F, and G and H, at the cost of 8 taken pieces.
Of course, that “white’s A pawn takes a black piece” is wasteful in the sense that it is both a pawn move and a piece being taken, but I think it still is better than letting the pawns end up facing each other without being able to move more.
http://tom7.org/chess/longest.pdf