Of course you have to admitt that the game has to be ended as soon this is possible according to the rules (if someone can claim the 50-moves rule to end the game, he has to), and that the question makes sense we will exclude the option of resigning
I think iamatiger is on the right way, but the I get a different result (a try here):
lets say that you are inventive enough to avoid the end of the game by the repetition rule (concurent repetitions or other never mind), then the limiting facter is the 50moves rule. So, as said by iamatiger we need either a pawn move or a capture to have a "reset move".
We find a maximum of 6*8 pawn moves on each side giving 96, and beeing able to capture anything except the kings we can have 30 captures. we can find therefore an upper bound: 99*(96+30)=12474 halfmoves or 6237 entire moves.
The king shuffle game adds another 99 halfmoves. At least one halfmove is lost lost during "resetmove color switching" (the 98th halfmove instead of the 99th resetmove must occur) and white has to begin (we don't want to lose a precious halfmove already at the beginning, do we?)
Leaves 6236,5 moves max.
a question is how we should manouver our pawns past the enemy pawn line - remember, if a pawn is lost, you lose all the resetmoves connected
to him too!
if the color is not switched (in terms of resetmoves) more than once this would be impossible - a black pawn must be captured in order to give the white ones a breach where to get through! so the art is manouvering the pawns through enemy lines without:
a) capturing pawns
b) switching colors as seldom as possible
I count the following color switches (of cours do your fillingmoves in between):
bringing the white pawns in position: 0
then the black ones: 1
get the first 4 white ones behind enemy lines (4 black pieces are captured = takes 4 resetmoves away, because a pawnresetmove and a capture move are the same here!): 1
now the now you switch and command 4 black pawns behind the opposing 4 white pawns (which are still in the line of advancement).:1
all pawns are free now, (another 4 pieces are captured by pawns) and we continue with the black pawns to promote them.
we switch again to white, making the last pawn moves and blasting any enemy pieces out of existance (just let the king live): 1
the black king naturally takes vengeance and does the same with white:1
takes in all 5 color switchings and 8*99(=792) halfmoves (for the pawns capturing pieces), thus leaving me with a funny calculation and a result of 11677 halfmoves, with the black king making the last move in the 5839th move in the game.