Interesting website: https://wismuth.com/chess/longest-game.html
At least it is up-to-date with regard to the FIDE-rules, using the 75-move automatic draw rule as decider for absolute game durations rather than the old 50-move claimed draw rule which leaves it up to the players to terminate or continue.
Well... games CAN go on forever. It's not mandatory that the game end after 50 moves without a pawn having moved or a piece captured. Yes, at that point either player can CLAIM a draw, but if neither player claims it, the game goes on. This is true in skittles games and over-the-board tournament chess as well.
However, let's rephrase the question: What's the maximum number of moves a game of chess might last before one side is able to CLAIM a draw?
This site:
http://www.chess-poster.com/english/notes_and_facts/did_you_know.htm
lists the answer as 5,949. But it doesn't back it up.
I think that once the two kings are left on the board, in theory they can also shuffle around for 50 moves before a draw is claimed. That might explain the difference between your answer and the above site's answer.
Ahhh I see what I did wrong! I assumed that the first move must be a pawn push, but in reality, you can wait 50 moves before the first pawn push. Thank you for that!
A draw will automatically be claimed by the arbiter with the 75-move rule