I'll limit my list to Playstation stuff or it might get too long
GT1 - I owed a Prelude at the time and the game got it PERFECT, the "tap, tap, tap" of the engine at idle and even the location of the key to unlock the trunk. 160 cars? No freakin way! Before that the most any racing game had was like maybe 20 (Ridge Racer?). GT was the game I had always dreamed of: take a normal every day car, race it, use the money to buy parts, tweak and tune then repeat till the car was a unstoppable monster.
Tomb Raider II - I think it was level 3: I got mad because I couldn't figure out how to reach a certain area, so I just started shooting EVERYTHING in sight... this broke out a glass window and I was jump to a rooftop and access all this other stuff. The underwater level with the upside-down ship was incredible: full of detail and the need to have a flare to see it all. The game seemed to be endless, when I finally finished it MONTHS later (after a 2AM marathon session) I left like I had really done something special.
GTA III - when you first stepped out onto the street from your safe house and watched a hooker & pimp get into a fight as newspapers blew by during a rain storm it was like a tiny world opened up right in front of your eyes. At times in that game I would just stop and WATCH the AI interactions... it was truely "next gen". I remember crashing cars just to watch people run from the scene screaming, then a fire truck would pull up to put out the blaze all without any input from the controller. No game had ever done such a sophisticated simualtion of the real world before.
Ratchet and Clank - leveling up the weapons just to see the effect they had on enemies, this made each level worth playing over and over. The way the designers made you return to previous levels with new gadgets (sling shot, mag boots, etc) was pure genius. The colors, the dialogue, the weapons, the enemies, the cutscenes, everything made it feel like a playable cartoon.
Burnout Paradise - the first time I went online in the demo and it was FLAWLESS, with no slow downs and full chat, all seemlessly intergrated. It was like "so this what how an online game should work". The way the cars broke apart, spun and flipped, yet felt totally controlable. The game seemed to egg you on to try insane things: do three spins off a bridge? No problem, watch this! The very defintion of fun.