Never actually had one myself, but a few of my friends had one, so I'd pick the N64. I was right at the age where I lived and breathed Pokemon, so Pokemon Stadium and the gameboy games were what I played whenever I could. Goldeneye was our go to game as well when we were all burnt out from playing Pokemon.
First ever console I owned all by myself was the original Playstation, so that's got to be up there too. That's when I first discovered Gran Turismo and continued to buy the GT games (PS2 as well) until I eventually had to decide between 360 and PS3, and went with the 360.
Genesis. Mostly for the same reason as AlphaWolf: sports games took a huge leap from double dribble and blades of steel to NHLPA and Lakers vs Celtics. The introduction of Sonic, Shining Force, Phantasy Star, arcade ports like Forgotten Worlds. For my money, the only loss Sega took during the 16-bit era was SF2, and quite obviously the Nintendo franchised titles.