Wow, my list got pretty long 
Anyway, here they are (I tried to include as many systems as I could still remember).
Half-Life 1 (PC) - This one is quite obvious. Not only did you get HL1, which was a great game in and of itself because it had a good story and it was just non-linear enough to make you feel like you had a choice, but you also got Team Fortress Classic (one of my all time favorite games), Counter-Strike, various single-player games that were quite good, and Day of Defeat.
Half-Life 2 (PC) - Again, a great game in and of itself but you also got DoD:S, CS:S, and independent mods like Gary's Mod (before he started selling it) and upcoming mods like Fortress Forever. Though I loved HL2, I think the HL2:EPX release model is crap so I don't include those games.
Mario RPG: Legend of the 7 stars (SNES) - I absolutely loved this game and played it multiple times through. It had all the familiar Nintendo characters and as an RPG it was actually fun and had a good story IMO.
Final Fantasy VII (PS1) - I loved the story and the gameplay, plain and simple. The materia system is my favorite FF system of all time because it allowed me to make my characters do whatever I wanted them to. Also the graphics and presentation (for its time) were so amazing.
Final Fantasy X (PS2) - The voice acting may have been total crap and the story may have been trite but the battles, mini-games, and new take on summons was still great in my book.
Bioshock (XB360) - I don't think I have to tell anybody why, but the story is amazing, the graphics are great, and the gameplay is a lot of fun.
Trauma Center: UtK (DS) - It may have been extremely hard, but this game had me coming back for more punishment time after time. Even though the surgery became completely abstract and unrelated to actual surgery, the gameplay was addicting and fun.
Super Mario World (SNES) - There's nothing like a great platformer, and Super Mario World was pretty much the platformer for me.
Gran Turismo 3 (PS2) - Even though I hate the license tests, GT3 was just accessible enough for somebody who isn't that great at racing sim games to be fun.
Kingdom Hearts (PS2) - Does not include the disappointing KH2. I first though KH was going to be a stupid game because of its concept (Disney characters?!) and yet it was so much fun and so engaging.
Pokemon Red/Blue (GB) - I played this game so much... There is just something about collecting and leveling up Pokemon that I love.
Final Fantasy Tactics Advance (GBA) - I know that a lot of people don't really like this game, but I really liked it. It was simple enough that I could play it in short bursts on the subway or at school between classes, yet it was deep enough to keep my interest the whole way through. The post game may be boring but the game is great overall.
Chrono Trigger (SNES) - One of the best RPGs of all time. I loved the new game+ and I loved that it had so many possible endings. Plus, the multi-character attacks were so cool. I can't believe this game hasn't been remade, or even ported!
Super Mario 64 (N64) - This is the first game I played on the N64 and I thought it was great. I spent so much time just collecting all the stars just for the bonus at the end.
