Here is how I found what I believe to be the largest chess game possible.
Remember the 50th move rule, which states that every 50 moves, a pawn
move or a capture must be made. This ensures that there is a finite number
of possible moves.

We will assume that black or white can move on the 50th move to make things 
simple. 4 of black's pawns will capture 4 white pieces (not white 
pawns), and 4 of white's pawns will capture 4 black pieces. This enables 16 
pawns to move 6 spaces each, and 16*6=96. How many pieces 
are left to be captured? 32 pieces -8 captured pieces=24. Actually, 22 pieces will be 
captured because the two kings can't be captured. BUT, when the two kings are left, they will 
move 50 additional moves until the game is a draw. So that is like "capturing" a 
king. 24 pieces - 1 king = 23 pieces.
Multiply 50 by the sum of 96 and 23, because 50 moves ensue before a capture or 
pawn move, and you get 119*50 which equals 5950. This is close to the number 
5949, which one website has posted.

This is not correct because we must now calculate how many times the 
captures/pawn moves SWITCH from black to white, and subtract that number from 
5950. Here's how. At first, 2 black pawns capture 2 white knights. Now, black can move 4 pieces 
(2 knights, a rook, and a bishop for example) onto squares where white pawns can 
capture them. In the meantime, white can move his rooks back and forth while 
black moves his knights around or develops his pieces through the openings that 
the black pawns created. This proves that both sides can move around while 
waiting for the 50th move. When white captures the black pieces, he moves on the 
49th move, so we subtract 1 from 5950 to get 5949. After white captures 4 black 
pieces, the white pawns can not all promote because two black pawns still need 
to capture white's pieces. So, we need to make another switch. 49 moves after 
white's pawn moved, black must make a pawn capture to get all of his pawns ready 
to promote. 5949-1=5948. With black moving
 on every 50th move, he can promote all of his pawns. He can't capture all of 
white's pawns and pieces because white still has to promote all of his pawns. So 
white must move; we need another switch. 5948-1=5947. Now, with white making a capture/pawn move every 50 
moves, he can now capture all of black's pieces except the black king. Note: 
white can position his knights (pawns can promote into knights) in order to 
shield the white rooks, bishops, and queen(s) from checkmating the black king 
while he is alone. We need one more switch; the black king must now capture all 
of the white pieces! 5947-1=5946.

After there are two kings on the board, they move 50 more times until the game 
is a draw. As you can see, most of the time, there are 50 move intervals between 
pawn moves/captures, whether they were made by white or black. There are only 
four 49 move intervals between pawn moves/captures, as I just showed, which is 
why I subtracted 4 from 5950. I simulated part of this game to explain below
(only around 20 moves, not 5000!). Please comment if you disagree or have an alternate
solution that gives more moves. I would like to know why someone calculated 5949 moves.
That is only 3 apart from what I calculated, so I'm sure our solutions are similar. Note that
this may not be the best solution; of all the solutions I could think of, this was the 
largest number of moves.

A slight fly-in-the-ointment is that the 50-move rule does not automatically force a draw - according to FIDE's rules. The relevant section is reproduced below.

10.12

The game is drawn when a player having the move claims a draw and demonstrates that at least [the last?] 50 consecutive moves have been made by each side without the capture of any piece and without the movement of any pawn. This number of 50 moves can be increased for certain positions, provided that this increase in number and these positions have been clearly announced by the organisers before the event starts.
[The claim then proceeds according to 10.13. The most extreme case yet known of a position which might take more than 50 moves to win is king, rook and bishop against king and two knights, which can run for 223 moves between captures!] http://www.chessvariants.org/fidelaws.html 

Thats a lot of moves

well that's not so bad :D that game will take you at LEAST 80-90 hours :D of course--that figure depends on the average move time.