O.D.T. - Escape or Die Trying: A rather obscure action-adventure that came out for PC, PSX and N64. It's pretty hard and unforgiving, but has some really awesome level and enemy design and an excellent soundtrack.
Pathologic: The less you know about this game before playing it, the better. Though this game has some pretty serious, mainly technical flaws and a pretty bad translation - something that will hopefully get adressed in the upcoming remake.
Cossacks - Back to War: Just simply my all-time favourite RTS out there, because it actually managed to pull off an economy system with undepletable ressources and a practically non-existant population limit - this game is for people that think SupCom's scale was too small, it's economy too simple and it's gameplay too close to StarCraft.
Homeworld 2 (and probably Remastered which comes out today on Steam): Close rival to Cossacks for my favourite RTS (but only because of the Complex Mod really).
Broken Helix: Another obscure PSX action-adventure, featuring Bruce Campbell, real-time dependant gameply in Area 51, 4 different paths through the game and a fucking button to switch your characters mood into ass-kicking mode.
Lethal Skies: Ace Combat's nerdy brother, striking just the right balance between simulation and arcade (man I really gotta get the second one at some point...).
Darkest Hour: Europe '44-'45/Red Orchestra: Just pretty much the most realistic platoon level WW2 FPS out there, featuring slow-paced tactical combat, combined arms and tank warfare with a huge variety of vehicles and great moddability and customization.
System Shock 2
Fallout 1,2, NV
Legacy of Kain series
Armoured Core series
Dungeon Keeper 2
Death to Spies series
Mother 3: Holy shit. Just... holy shit. Talk about an emotional experience. Finishing this game was such a ride I was stunned silent for fifteen minutes. From beginning to end it's probably the best example of how to play to emotions using a video game format. The entire series is brilliant, but Mother 3 takes it to a whole new level. (Really the whole series is here, but Mother 3 gets the mention)
Silent Hill 2: Yeah, it's been brought up in this thread a lot, for good reason. I can't really think of anything to add. Everyone's already said it. Honourable mention from the series goes to Silent Hill: Shattered Memories. Sure it may not be scary, but god DAMN that story and psych profiling.
Super Metroid: If I had to tell someone what I thought was objectively the best game of all time, I'd probably point to this. I can name a million things wrong with all of the other games in my favourites, but I seriously can't think of a significant flaw in this game. Controls are tight, atmosphere and art direction are fantastic, everything feels balanced and just plain fun. I could replay this game forever. (the only reason Metroid Prime isn't here is that thanks to my motion sickness I can't play more than 15 minutes of Prime at a time. It's a shame. I love it, too)
Pokemon: It's true. I've been playing Pokemon ever since it landed on Western shores and I'm still not sick of it. I could probably replay any given Pokemon game forever because I love the little bastards so much, all of them. Though if I had to choose a favourite generation it would be 5 or 6.
Sonic Adventure: Yeah, yeah, it's a Sonic game. But out of the entire franchise, I think I enjoy this one the most, even though it's not technically the best. This was the last Sonic game to really feel like the classics until Colors/Generations while bringing its own innovations, and the Adventure stages made it feel surprisingly immersive. I could replay this game a hundred times.
Team Fortress 2: For both sentimental reasons and because it's an awesome and fun FPS in general. It's thanks to this game that I met my spouse. What isn't to love? The gameplay is fun, the stories and characters are fucking hilarious, and everything about it is great.
Honourable Mentions: Animal Crossing: New Leaf, Spore, Minecraft, Legend of Zelda: Twilight Princess/Wind Waker, Super Mario RPG, Mega Man 2. I'm probably forgetting some but that's what I can think of right now.
Arkham Asylum. For a re-introduction of Batman into the vidya, this game was near perfect. I loved the actual detective feel of the game. The combat was lacking, but the Riddler challenges made up the difference. This was the first game I really played on my own outside of FPS.
TES: Skyrim. I agree that the plot lines were lacking in some places, but the sheer volume of the game and the beauty of it is fucking ridiculous for a 360 game. Dragons.
LA Noire. I love the period and I love detective games. Similarly to Skyrim, the intricate detail of the game really gets me every time.
Assassin's Creed: Black Flag. The introduction of seafaring and raids made the game way more interesting than any other AC game. I actually cared about the plot and the characters and started fucking crying at the end like a babby.
Planescape: Torment- There isn't really anything I can say about this. I love it. So much. The gameplay is a bit wonky at times, especially the combat mechanics (Seriously, fuck whoever designed Curst prison), but come on, you're not going to be playing it for those things.
Shadow of the Colossus- Both stunningly beautiful and balls-out fun.
I totally didn't cry when Agro fell from the bridge, and there's nothing you can do to prove otherwise. I also didn't cry when she turned out to have survived. So there.
Spec Ops: The Line- Anti-war without necessarily being anti-soldier. Made me realize that having the best intentions is nowhere near enough when dealing with a problem. Also that I'm probably not someone who should be allowed access to firearms. Hooray for England's gun laws.
The Legend of Zelda: Majora's Mask- I've never understood why Ocarina of Time tends to be held up as a shining example of the series over this. It gave me recurrent nightmares of impending apocalypse and giant faces in the sky for months. And that's a good thing.
The Thief Trilogy- A perfect example of how a trilogy is supposed to work; first you examine one side of a theme, then the other, and then you deconstruct the whole thing and tie it all up into bookends. Weirdly, a great deal of my attitude to politics and social movements comes from these games, namely that becoming partisan in anything can be dangerous.
The Cat Lady- For someone who's dealt with depression, this hit me right in the feels. It's incredibly bleak in aesthetic, but is ultimately rather uplifting in outlook. Also passes the Bechdel Test with flying colours, if that's important to you.
Honourable Mentions- LoZ: Wind Waker (Nearly as good as MM), Bastion, Blood, Sherlock Holmes: The Awakened (Holmes and Cthulhu, fuckin' A), Okami, Forbidden Siren, La Mulana, Painkiller, Broken Sword, Gabriel Knight, Betrayer (Quite new and under the radar), Dark Souls.
Super Smash Bros. Melee is my favorite game of all time, gameplay wise. It's really competitive and I love it.
EarthBound is one of my favorite game when it comes to atmosphere and easy going RPG's
Majoras Mask and Ocarina of Time are tied for my favorite game to play through a few times per year.
Journey - If you haven't played it, find a PS3 with an Internet connection. My first time through was one of the most emotional experiences I've had with games.
Ratchet and Clank - I can't pick a favorite of these but I can narrow it down to Crack in Time or Up Your Arsenal. Inventive weapons, funny universe, excited for the movie in 2015.
Jak and Daxter: The Precursor Legacy - Before the series got edgy (don't get me wrong I love Jak 3 too), it was a fun, creative platformer with eco powers. Nice humor, cliffhanger ending sucked, but definitely my favorite platformer.
Starcraft II - I hit Masters rank 16.... that game consumed me for a whole 2 years. It's good but.... I mean it showed me PC gaming.
Dawn of War is one of my favorite series. I thought the gameplay was fantastic and I've poured hours into the series. A lot of people thought the writing was bad but it was one of my favorite parts. Soulstorm was a literal abortion though.