I usually binge play a game for a month or 2 then move on. Angry Birds I played and played until I beat every level and 3 stars on many. Never got sick of it, though often frustrated. Bloon TD I have been playing for a few months, and still haven't gotten sick of it. Great game.
I am sure I am missing some, I have played a lot of games over the years. Lets hear some of yours!