The number of simultaneous games you can start is currently 200 games. If the limits is reached you get an error when you want to join a game or invite another player to a game. Other players get an error as well if they want to invite you to more games if your limit is reached.
 
If 75% of the maximum number of games is reached a warning is given when you want to start new games. We advise you not to start too much games, because then you have no reserves. It's also easy to loose track when there are too many games at once.