There are some positions in which you need to know how many moves have been played without pawn move or capture. For example:
[FEN "1R5K/8/8/8/8/8/7R/k7 w - - 98 1000"]
(1R5K/8/8/8/8/8/7R/k7 w - - 98 1000)
Here, white must play Ra2 or Rb1 to force a capture within 1 move. If this is not done the game is drawn due to the 50 move rule.
If the 50 move rule was not a concern it would be an easy mate in 2.
Of course it is very hard to remember how many moves have passed. Is white allowed to look at the scoresheet and count the number of moves during the game?