1. 9 Hours, 9 Persons, 9 Doors (12/10/09, DS)
Absolutely mind-blowing at the time, and still one of the best and most engaging mysteries out there. The localization leaves a little to be desired, especially compared to later games in the series, but comical overuse of "what the hell" is a small price to pay for how amazing everything else is.
2. Mass Effect (11/20/07, PC)
For my money the best introductory world building in any game. It's ugly and the gameplay hasn't aged especially well, but I've never felt so apart of a game's world. Twelve years later, there are still only a handful of games that have even gotten close.
3. Persona 4 (7/10/08, PS2)
Persona 4 did for a rural Japanese town what Mass Effect did for a galaxy. It's got dungeon crawling and world saving and that's all great, sure, but its biggest accomplishment is making day-to-day rural life so believable and relatable. Like ME, I've finished it four times and still haven't skipped a line of dialogue. P3 could easily have been here for all the same reasons except that the combat is crap and P3P didn't fix it until 2010.
4. Pokemon Emerald (9/16/04, GBA)
It's just a really good Pokemon game, and you've all played it. It had Rayquaza and Flygon and a ton of others and I actually love the horns.
5. Mirror's Edge (1/12/09, PC)
Far and away the game I've completed the most times, despite the fact a lot of it is linear and the story is truly awful. It makes up for all that by doing parkour really well, and turning what would otherwise be a dry and forgettable campaign into a nearly unbroken series of pathfinding and movement challenges that have you pulling off tricks games normally reserve for cutscenes once you master them.
6. The Legend of Zelda: The Minish Cap (11/4/04, GBA)
This could easily have been Wind Waker, but MC won out because the Triforce quest is awful and MC is consistent the whole way through. And that's also what makes it stand out in general: there aren't really any moments or mechanics that make MC brilliant on their own, but the way the whole game fits together so that the main line is a constant push of new content while still always having loads of worthwhile side stuff if you want it is everything I want from Zelda.
7. Silent Hill 3 (5/23/03, PS2)
It's a great survival horror game in its own right, but what really elevated it for me was the contrast between how seriously all of the supporting cast takes everything and how the protagonist just can't be bothered with any of this supernatural nonsense for large stretches of the game. And it's got the UFO ending!
8. Resident Evil 4 (1/11/05, GC)
Pretty much the same thing except that now the protagonist is a rejected action hero and spends the whole game making bad one-liners while ridiculous vaguely-horror things happen. It isn't very scary, but it's great comedy.
9. Left 4 Dead 2 (11/17/09, PC)
Still one of the best co-op experiences out there because of how well the level and enemy design fit together and how the AI-controlled spawns keep things just a little different every time you play. It's also the third horror game that's really a comedy, with silly goals like escaping with a garden gnome and pointless carnival mini games.
10. Final Fantasy X (7/19/01, PS2)
The last good one. Great music and story, clever ways of making leveling up and turn order interesting after 10 games, and it even had Blitzball.
Unordered honorable mentions: