Well, I feel that the following 5 games represent me very well:
1. Sonic the Hedgehog 2
Yes, not Sonic 1, not Sonic & Knuckles, the first video game I ever owned, but Sonic 2. Why? Because aside from being the first Sonic title that I have ever played, it was also the first video game that got me interested in video games to the point where I wanted to buy a console and start owning games. I have enjoyed the NES, SNES, and various arcade titles, but none of them got me excited about video games as Sonic 2 did.
2. Final Fantasy X
Why not any preceding and "debatable" better numbered titles? Because I owned a PS2 and Final Fantasy X was the first FF game on the PS2. It was also the first RPG to get me interested in other RPGs, making it one of my favorite genres.
3. Devil May Cry
I'm not a huge Devil May Cry fan, and I've only played DMC 1, 2, and 4, but DMC was the game that helped me get desensitized by all the mature content and gore that is common ground in this day and age. Before DMC, I couldn't play Jurassic Park for the SNES, Onimusha on the PS2, etc. Being able to deal some serious damage to those Marionettes in style gave me the confidence to press on and only deal more seriously stylish carnage.
4. Bioshock
Why not SystemShock 2? Again, Bioshock came first to me. This was the first title that proved to me that a FPS can be more than dull, repetitive, DLC spewing, multiplayer death matches, and could actually provide an enthralling single player campaing, a great story, a daunting atmosphere, customization to fit you're play style, and having options on how to dispose of your foes. After Bioshock, I've kept an open mind to many FPS and Third-Person Shooters, both obscure and mainstream.
5. Tetris
I'm good at Tetris, plain and simple. Everyone has that one game, be it casual or complex, that they are inherently good at. They can play it for weeks like a champ, put it down for several months, and pick it back up as if a day hasn't gone by without them playing it.