Probably Final Fantasy 9, I just grew up with it as a kid and the story had a big impact on me. And the Devil May cry series.
Too many games to list Battlefield 1942, Half-Life, Counter-Strike, Max Payne, Zelda Ocarina of Time...
League of Legends is a great free game to play. It's competitive, free to play, and is overall a great strategy game. It's based of DoTA if you've ever played that in Warcraft III. (I'm actually in a league of legends game right now lol alt tabbed).
May be it's not popular now, but my favourite is first half-life. I like old games such as quake2, unreal/unreal tournament first.