I can't say with 100% certainty, but I'm pretty sure that my first single-player domination was Super Mario 64.
Multiplayer ... Well, it's hard. Nowadays, I do that with pretty much any fighting game, as well as some other multiplayer-things ... I think the first one I really dominated in multiplayer was Mario Kart 64.
I gamed a lot before the N64, but I think the N64 was when I first really found my own.