The maximum number of moves in one game that a player can make is somewhere between 5,871 and 6,300. See "The Longest Possible Chess Game" for an explanation and intelligent discussion of this question.
As far as how many can be made on this server, why don't you tell us what your own research has revealed?
I think that this is not okay. I played a game where the opponent was giving me mates with his queen for 30-50 moves without stopping. He didn't want a draw and I had to quit at move about 130, because at that moment it stopped being a chess game, it became nonsense. I appeal this to be stopped.
Should 3 repeated moves be the maximum, in my game there were 4 and nothing happend so we moved on?
This site follows the rules of chess, which state that you can claim a draw once the same position occurs for the third time. But it's not automatic, you have to click the draw button for it to happen.
What is the maximum number of moves in one game that a player can make in this server?
What is the maximum number of repeated moves?