Okay here it is. I was thinking about this for ages... and to be honest, if you asked me again tomorrow I'd probably change my mind on some of these, but as I had to come up with it from the top of my head, these are what I pulled out.
Honourable mentions: Persona 4, Mass Effect, Super Mario 64, Super Mario Galaxy, Ocarina of Time, Pokemon Snap, Pokemon Blue, Civilisation IV... Probably some others.
Okay, explanation time
Killer 7 - I adored this game. I can't remember what made me pick it up, but it was just amazing. It's my favourite game of all time. It was so strange but it's political nature made me really think about it... The characters were great, finding out how they became the Killer7, and how they came to possess the abilities they have, was mindblowing.
Final Fantasy IX - It was my first Final Fantasy game, and most people enjoy their first one more right? But I do think there's more to it than that. When all the recent Final Fantasy games had miserable, emo heroes who didn't much care about their companions, along came Zidane, with his happy-go-lucky attitude and kind caring personality for his companions, it was a breath of fresh air. Not to mention, Vivi is the most badass mage in the series.
Phoenix Wright: Ace Attorney - Man, this was difficult. Of all the Ace Attorney games, I had to pick just one to put in the 3x3... In the end I settled for the first one just because it was the first one to do what it did, but you can take it to mean that I'm talking about the whole series here, or at least the Phoenix Wright arc. The stories these games spun were awesome, the characters were so numerous, and yet so memorable, that I could probably name most of them in this post, even the ones that only appeared once.
Metal Gear Solid 3: Snake Eater - I love the Metal Gear Solid series... I had trouble deciding between this one and the two the preceded it, but in the end I settled for MGS 3 because of the very raw, personal story that wasn't really apparent in the other games. While others had huge, sweeping stories of nuclear warfare and the patriots and what have you, this one, which still contained all that in itself, was more, at its core, a story about a man, and it was an amazing story, perhaps the first game to draw tears from me at the end.
Broken Sword: The Shadow of the Templars - Not a common choice, I know, but this was my first adventure game and it was awesome. I loved the story, I loved the main character, Geroge Stobart. It's a shame they degraded in quality as the series went on, but I remember the first two in the series being simply great. This was before I had played games like Monkey Island and Sam and Max.
The World Ends With You - This was and in my opinion, still is the freshest RPG to come out of Japan in a long, long time. The story was great, with plot twists and complications to keep me playing. It was interesting, it did something new, the battle system was great fun once you got the hang of it.
Crash Bandicoot - One of the strongest gaming memories I have as a child was of me finally completing this game, defeating Cortex, saving Crash's sister... It was platforming greatness.
Kingdom Hearts - The first game that I liked enough to join a forum about it. I loved the whole game, the story was great, the battle system was fun, the disney and Final Fantasy cameos delighted me.
Fallout 2 - I'm one of those. Yes, those who played Fallout 3 before any of the other Fallout games. Then I played Fallout 1. And then I played Fallout 2. There are a couple of reasons Fallout 2 comes out on top for me. Firstly, there wasn't a time limit, so I could dick around in the world as long as I liked, which I loved. Secondly, the dirtier, seedier world of Fallout became apparent. It had great references and humour, even if that did piss a few people off, and I just loved it.