Hmm, probably best if I list mine per genre. I'll try to think and keep only 1 per category.
RPG - Chrono Trigger
Great game made for the SNES with a followup sequal called Chrono Cross, which I could never find to buy as it was only a US/Japanese release. Chrono Trigger basicaaly is a game about time travel and is a turn-based RPG. The story itself is amazing and will keep you gripped, and the characters have their own individual personalities as well which is good.
Other games worth a mention in this category: Star Wars - Knights of the Old Repulic, Baldur's Gate, Silver and Golden Sun.
FPS - Call of Duty
I know a lot of people will say "CoD, Really?, why not Half Life 2?" Well to be honest although Half-Life 2 and Counterstrike are good games, I just loved Call of Duty, both the single player and even more so the multiplayer. There were close seconds such as Unreal Tournament, but Call of Duty was about the first FPS I really got addicted to, I think I played it almost constantly one summer.
Other games worth a mention in this category: Counterstrike, Half Life 2, Unreal Tournament, Doom.
Sport - Virtua Tennis
Highly addictive game on the Dreamcast (I think it was also on PS2), I could sit and play for ages because it was so easy to play, yet challenging to beat opponents as you progresssed.
Other games worth a mention in this category: Need for Speed Underground
Fighting - Dragonball Z Budokai
I've played many game sin this genre including Street Fighter, Capcom Vs SNK and Soul Calibur, but I haven't really enjoyed any of them. Budokai was a mixture of nicely combined special moves, with cartoon graphics. Very fun and about the only game in this genre I like.
Other games worth a mention in this category: None
Adventure - Monkey Island 2
I could talk for hours about how this game got everything spot on, and why it is better than all the other Monkey Islands, and all the other adventure games. What I really liked about this was that it kept everything witty, had a unique story-line and was piratey. There was also the free roaming aspect of travelling to any of the three islands at will, kinda of like a early version of GTA gone Caribbean.
Other games worth a mention in this category: Beneath a Steel Sky, Sam and Max, Broken Sword and Grim Fandango.
Simulation - Rollercoaster Tycoon
This game was great, I've played things like RCT 2 and 3, but 1 was the best and most addictive for me. There were similar things such as Theme Park, but had very limited ride construction tools. I've also played many god sims such as Black and White and Settlers, but I enjoy building rollercoasters and RCT is the most fun I've had from any simulation game.
Other games worth a mention in this category: Settlers 3, Black and White, Theme Hospital and Sim City 4.