Longest Possible Game

First of all, in chess.com, it will not auto-draw the game until 75 moves have been made without a pawn move or capture.

Next we should be clear what "move" means in chess. A "move" is 1 move by white plus one move by black. So for example, at the start of a max move game the players will shuffle their knights around avoiding a 5 fold repetition (that's right, chess.com doesn't auto claim until 5 fold repetition). And after white makes his 75th move, the game is not yet automatically draw because remember black has to make his 75th move too. 

So right away we see that every pawn move, and every capture, represents the 74.5 moves that came before it. The last move of the game will end with K vs K, and that last capture will also represent the 74.5 moves that came before it. So now we have an easy way to count.

Naively, we simply count the maximum number of pawn moves (96) and the maximum number of captures (30) and then multiply them by 75 (126x75=9450).

But there are two factors to account for that reduce this number.

1) First is that pawns are blocked by enemy pawns. A player can't make 48 pawn moves straight down the board without capturing to move around the enemy pawns. So some moves are pawn moves and captures at the same time, which means we double-counted. With a little thinking, you can figure out that only 8 moves have to be both a pawn move and a capture at the same time. So that brings it down to 118x75=8850. If you got this far on your own (or the 50 move equivalent which is 5900) then you did really well! But there's another trick to notice.

2) Remember at the start of the game when both players shuffled their knights until white had made his 75h move? And now black makes a pawn move (or capture) to avoid the auto-draw? Well just think... black can't always be the one to avoid this, because eventually he will run out of pawn moves (or captures).

So our second factor is noticing that every time we have to switch sides like this (from black's responsibility to make the draw-avoiding move to white) we lose 1/2 move.

We note that sides must "switch" when:
1) The only thing left to capture of the opponent's chessmen are pawns.
2) a pawn can no longer advance

When a player can't capture or move a pawn, then the responsibility must switch. We can do it more often, but keeping it to a minimum will make the game as long as possible.

Again with a little trickery, we come up with we only need to switch 3 times! (If you thought this number was 4, you were really close!) So now we calculate 1/2 move x 3 is 1.5 moves.

So we have (118 x 75) -1.5 = 8848.5 moves. (For those of you just reading bold text, this calculation is for chess.com's 75 move draw rule)

In other words the game will end on white's 8849th move, and with just K vs K left black will never get to make his 8849th move.

For nearly 75 moves the players can just shuffle around, but then a player must make a pawn move or capture to avoid the draw. That move will be the only precise move for that set of 75.

Black has the first 12 precise moves. He uses these moves to make that pawn structure (which involves capturing two white knights).

White has the next 49 precise moves. (He uses these moves to promote all his pawns, and capture all of black's pieces.) Note that black's pawn structure stays the same because during this phase white is the one responsible for making pawn moves and captures.

Black has the next 49 precise moves. He uses these to promote all his pawns and capture all of white's pieces. Black should be sure to promote one of the pawns to a queen or rook. This queen or rook will be the last piece captured.

White has the next 8 precise moves. He will use these moves to capture all of black's promoted pieces, ending with a queen or rook.

The diagram below is right before white makes his 8849th move which will end the game.

And here's what it would look like in full form.

Remember in the game below, every pawn move and capture represents the 74.5 moves that came before it.