Sonic 2 and Mortal Kombat 2 are the games I remember most from my childhood.
CTR was the game that came with my Playstation and I just remember sinking hours upon hours into it as well as it being one of the few games my Dad would willingly play.
Diablo 2; I played it for the funsies and it lead me to Diablo 3; I play it because I'm compelled to by some horrible unseen force that won't let me stop until my mouse finger has seized into a horrible disfigured claw.
FF8 was the first Final Fantasy and probably JRPG that I played, not really my favourite but I guess it's significant.
Madden 2003 was the first sports game I really got into, which is a big deal because almost every year I'll buy Madden, NBA2K and MLB the show.
Final Fantasy X was the first time I was ever blown away by what I was looking at coming from my game console. The graphics actually took my breath away and I was hooked. Probably the only FF I'm compelled to replay these days.

Kingdom Hearts. This series is getting its own little paragraph. I picked this game up completely on a whim, I saw on the cover it had Disney characters so I was worried that it might be for little kids (I was like 14 or something *eyeroll*). Anyway, point being that I have no idea what drew me to the game. I hadn't read write-ups or previews, I had only played FF8 and 10 at that point so I wasn't aware of most of the tie-ins. So either way, that was the game I chose and straight away the intro cinematic drew me in, then came my favourite tutorial to any game ever. I don't know why I love the sequence on the large platforms but even to this day when I play it picking between the sword, shield and wand fills me with excitement. More importantly the passage at the end gives me goosebumps even thinking about it ("the closer you get to the light the greater your shadow becomes... etc). Anyway, I'm rambling, that game made me laugh, cry, rage quit multiple times only to ultimately come back more determined than ever, and is the pillar by which I compare all other games. I could make an equally long write up about KH2 but I'll spare you, I just smurfing love that series. Also looking for a walk through and discussion about this game is the reason I joined EoFF where I met so many friends and the love of my life... Sooo yeah, cheers Kingdom Hearts.