I would get use out of a feature that would send you an email with a game link whenever an opponent moves in a particular game.  There could be a check box near each game board (or something) to indicate that you want notifications for this particular game.

This feature would be very useful for several reasons and you would like to know as soon as possible when you opponent makes a move:

1) When you have a game that is running low on time
2) If you a playing a particularly interesting game.
3) If others in a tournament are waiting for the game to complete.

Thoughts?