That was also my intuition. Unless there's a rule that can stop two immortal but dumb-as-bricks players from indefinitely cycling through the same non-capturing moves surely the answer is 'infinity'.
Well there is. The three/five fold rule. And 50 moves rule.
It depends what rules you're using, but there are the three-fold repetition and 50-move rules which allow a player to force the game to end in a draw. The catch is they both require one of the players to claim a draw under the rule, otherwise they can keep playing.
There is additionally the 75-move rule where the the game is forced to be over without either player claiming the rule (the arbiter just ends the game), that rule would give an upper bound regardless of the players knowledge of the rules.