I'm always going on about this, but Russ did mention in another thread that nagging might have some effect, so...
Games that complete when not on your move are not obvious. As well as wondering 'when will person X hury up and move?', you often miss final in-game messages. Yes, these games do show up on the 'My games page', but unless you (a) visit that page regularly and (b) archive your games a lot, it isn't an ideal GUI.
I tend to play the games waiting, visit the forums and leave. Rarely do I ever see the My games page. I'd guess that is the most common GUI mode of use, (except for the forum bit).
So what's my solution? Errmm....
Well it is an extention to the present scheme of X games waiting, Y messages: add a third item games completed. A completed game would be one where:
You just got check mated
Your opponent resigned
Your opponent accepted a draw
Your opponent was auto timed-out
Your opponent was booted from the site
Click on the message and it will run you through the final board positions, messages etc. of the finished games.
And if you really don't care to know: then enable/disable the games completed message in your site settings.