this topic is going to be open ended. I'm gonna list some well known games with a developed competitive scene, but you can leave any off if you haven't played them or add any that you feel I missed.
Quality can be whatever you want it to mean. How much fun you have playing it, how long it has lasted in your competitive circles, whatever.
Warcraft III
Starcraft II
Street Fighter III: Third Strike
Super Street Fighter IV: Arcade Edition 2012
Super Smash Bros Melee
Super Smash Bros Brawl
Counter-Strike
DOTA
League of Legends
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http://www.youtube.com/watch?v=npHDxSvwCE0