A very interesting question. Now a day it is probably quite a few
a completely new game

what is the average number of moves after which a game has never been played before?
It's probably impossible to quantify, since there are so many games played but in practice I'm guessing it's under 20 for sure. In looking at my chess.com games, using the most popular move on each ply, 15 moves got to a unique game. That was on Daily where using a book is allowed.
For my OTB games database, that was 6-7 moves (though I'm missing a few games).
In both instances I only have a couple thousand total games though.
For higher rated players, the average may be higher.
what is the average number of moves after which a game has never been played before?