As stated above, determining what a "classic" is or is not is a somewhat difficult task. However, I'll take a stab at it, listing titles I think have a highly unique atmosphere and/or reach a level in their respective genre that will hardly be reached again and/or contain gameplay that will be easy to pick up and play years from now, in the form that games like Contra or Mario feature.
Ninja Gaiden - One of the greatest action games to grace a home console.
Halo: Combat Evolved - A decent yet flawed single player with one of the greatest multiplayer modes in gaming history. Classic Blood Gulch will always keep me coming back to this.
Halo 2 - From 2V2 CTF to massive showdowns online; the glorious multiplayer here will have me and my friends coming back to this title in the near future and beyond.
Star Wars: Knights of the Old Republic - Transcending a lot of RPGs this gen, KOTOR gave you the
feeling that you really were in the Star Wars universe, as well as a decent if cliched story to work through. Oh, the replay value wasn't too bad, either. Hopefully Bioware will at least be thinking of making another.
Jet Set Radio Future - No explanation needed.
Gradius V - I heart you, Treasure!
Metal Gear Solid 3 - One of the greatest games ever made, although a bit excessive in form.
Final Fantasy X - Only one other Final Fantasy title - Final Fantasy VI (SNES) - is as polished as this one.
ICO - Beautiful. Simply beautiful.
Devil May Cry - Capcom for the win!
Devil May Cry 3 - Ditto.
Katamari Damacy - A breath of fresh air.
Virtua Fighter 4 - The best fighting game this gen, IMO.
Sly 2 - A sadly underrated and highly enjoyable game. A diamond in the rough, for sure.
Resident Evil 4 - Shinji Mikami proves once again why he's one of the best in the business.
Pikmin 2 - Miyamoto magic. It all comes together perfectly and is just a joy to play.
Legend of Zelda: Wind Waker - Flawed, but enjoyable, and of course: its Zelda.
F-Zero GX - F-Zero is one of the near flawless racing series out there. Every title since the SNES original are just damn great to play. Superb visuals and gameplay come together to create one of the greatest experiences in gaming.
Wave Race: Blue Storm - Nothing really can be said. It just falls somewhere in classic territory IMO.
Metroid Prime - It shouldn't have happened. Retro Studios, a train wreck of a development house should not have made one of the finest achievments in 3D gaming. Perhaps only the original Metroid compares to this.
Super Smash Bros. Melee - Addicting, fun and immersive. One of the greatest sequels ever.
Ikaruga (DC Port...) - A personal fave, right up there with Blazing Lazers and co.
The Legend of Zelda: Four Swords Adventures - One word: Fun.
Prince of Persia: Sands of Time - Probably the greatest platformer this generation. Ubisoft are really capable of makings some incredible titles, but this one just hit that high note so perfectly...
Burnout 3: Takedown - Perhaps the greatest arcade racer this gen.
Grand Theft Auto 3/Vice City/ San Andreas - Ignoring the controversy, they're fine games with some fantastic depth and execution.
Viewtiful Joe - Henshin-A-Go-Go, Baby!
Phantasy Star Online - Although I still want Phantasy Star V, this was a damn good substitute.
Super Monkey Ball - Another breath of fresh air, although I'm not too sure how well it'll hold up...
Rez - Sublime.
Killer7 - Style over substance, yes. But that was
some style and it did feature a highly unique atmosphere...
Psychonauts - Fan-fucking-tastic.