I'll try and cover every genre, and only one game per franchise, maybe a reason if controversial, here goes;
Crono Trigger, widely known as the best RPG ever
Final Fantasy VI, better than VII and X only due to depth of characters and story
Goldeneye, huge favourite for everyone
Half Life 2, better than 1 marginally
Ico, beautiful, masterpiece, that sort of thing.
Metal Gear Solid 3, not as interesting as 1, 2 or 4 storywise, the gameplay is something of a high point.
Phoenix Wright: Trials and Tribulations, an almost perfect text adventure and incredible fun.
Prince of Persia, the new one...it's gorgeous and gets everything right apart from collecting light seeds
Resident Evil 4, generally brilliant
Silent Hill 2, ask Yahtzee, I agree with him on this game
Sonic The Hedgehog 2, amazing fun and surprisingly challenging
Street Fighter II, the best beat em up
Super Mario World, read Sonic 2
The Elder Scrolls IV: Oblivion, this is the game with the most memorable moment ever for me, it was night and I was sneaking through a field shadowing someone, I looked up at the sky, I saw the stars, the grass looked as if it were glowing, then the sun rose and the world exploded into colour...honestly, it was spell bounding, I was in awe.
The Legend of Zelda: The Wind Waker, I could decide between Link to the Past or Ocarina of Time and then remembered how beautiful this game was and how much detail was in it, Link's eyes moving got me, everything about Wind Waker was magical just as a nintendo title should be.
Honourable mentions to;
Ape Escape, best use of the analogue sticks
Dissidia: Final Fantasy, what? It's amazing and so obviously the future of beat em ups
Far Cry 2, huge, deep shooter
Final Fantasy VII, it's 3D and great
Final Fantasy X, HUGE amount of detail and almost every character in Spira has a voice
Katamari Damacy, second best use of the analogue sticks ever
Metroid Prime, amazing
Mirror's Edge, one of the most thrilling atmospheric action games I have ever played.
Okami, beautiful
Pokemon Gold/Silver, Real Time RPG...just awsome
Portal, GOD!
Shadow of the Colossus, amazing
The Legend of Zelda: Link to the Past, almost perfect
The Legend of Zelda: Ocarina of Time , see FFVII