The more I consider this, the more moments come to me:
Street Fighter II: How I loved this game. I played it and played it and played it. Perhaps this was because I only had 3 SNES games at the time, but that's beside the point. My best moment from this was probably beating the game on the hardest difficulty with every character so I could see every ending. It felt like an enormous achievement at the time.
Final Fantasy VII: My first JRPG, and how I loved it. Indeed, how we all loved it. I bought this game on release and so did many of my friends. My moment isn't from the game itself, but the chat that circulated around it. Each day I arrived at work after an evening of playing the game, and I discussed it with friends that were doing exactly the same, about how cool it was, what disc we were on, and so on. Most people claimed, at some point, that they'd completed it, or defeated one of the harder bosses. Most lied. Good times.
Killzone 3: The game itself, for all it was pretty, was largely mediocre. However, it did have an enormously awesome gaming moment for me: it was the first game to convice me motion controls were worthwhile. I'd owned a Wii for quite some time at the point when I played it, and I'd been largely disappointed with the machine. The motion controls were mostly a gimmick or stapled on, and the few games that used them well were mostly mini-games and not very appealing to me. Killzone 3 changed my opinion on this. Once I got used to using Move on it, it was a revelation. So accurate. So fast. So awesome. Convert, right here.
Populous: I am a God! Of yes! I played this original God sim a lot. Picking a moment from it is hard, but it is probably lowering the land and watching people drown in the water. Sadistic and endlessly fun, and an important precursor to running every bastard over in GTA and shooting whores for the money you just paid them. Actually, on that...
Grand Theft Auto 3: This game and its open sandbox was revolutionary at the time. I cannot describe how awesome it was, and how fresh it felt it me. It was the incredible excitement this game and its spin-offs elicited that was the sole reason GT4 was such a let down for me, as it didn't feel like a big enough step up from the incredible games it followed on from. My best moment from GT3 is simply driving around and exploring in a fully realised 3D city, doing whatever I wanted. It was just mind-blowing how big and how awesome the game was.
Mass Effect: SPOILER ALERT. LET ME SHEPHERD YOU AWAY FROM THIS PARAGRAPH IF YOU HAVE NOT COMPLETED THE GAME. This deeply flawed masterpiece has one of my favourite game stories of all time, and that alone saw me through to completing it not once, or twice, but five times so far on two different platforms (and seven times for the sequel on three platforms). The stand out moment for me, beyond all the cool twists and revelations of the plot, is climbing the occupied Citadel tower after it has been siezed by Saren's Geth and watching Sovereign tear everything apart above. On my first playthrough it was a real WOW moment, and I still love it today.
The Eye of Judgement: Screw the game, it was the cards that fascinated me. It was my first foray into augmented reality, and watching little monsters sit on little cards on my TV was just so futuristic and 'next gen'. My eldest daughter, in particular, loved it. She's pick cards up and move them towards and away from the camera, watching as the creatures moved with them, and it looked like magic. Yes, the game was a pain, but some of the ideas in it were just brilliant.
More will come.