Super Mario Bros. 3 - Accidentally figuring out how to fly (rental game, no manual, no other exposure).
Pokemon Blue - The first time I used a link cable to battle a friend.
Gears of War - My first exposure to HD at a friend's house. Mind = blown.
Assassin's Creed - This was the first title I played after getting an HDTV, and the visuals hit home for me.
TIE Fighter - The Darth Vader missions, in particular, but the whole game -- dynamic music, awesome gameplay, great visuals, and the storyline/scripting.
Xenoblade Chronicles - The entire introductory chapter.
Dark Age of Camelot - The first time I went on a relic raid.
Shadow of the Colossus - The first Colossus was amazing, considering I went into this game completely blind at a friend's recommendation, but I had my "THIS IS AMAZING" moment with the third Colossus.
Guitar Hero II/Rock Band - The 'feeling' of playing these games hooked me for life.
Unreal Tournament - Definitely Facing Worlds.
Morrowind - The way the beginning was set up was definitely fantastic.
DOOM - The first time I played an 'online' game against someone was with this title, via dial-up.
Battlefield 1942/2142 - The first games I jumped into in these were just amazing. 1942 had the chaos of EVERYTHING on Wake Island, and 2142 I jumped in and picked up some guy's kit and suddenly had wallhacks (heartbeat sensor) and a turret -- I rolled Support for life.
MAG - Same sort of impact as Battlefield had on me, except with way more players, but I sucked at it (I suck at console shooters) so I never stuck with it.
Soul Calibur - One of the first arcade machines to absolutely blow my mind with the combinations of visuals, gameplay, and dat soundtrack.
MechWarrior 2 - I'm glad K.Jack said the first time they fired a salvo of LRMs, because I am right there with them.
Star Fox 64 3D - Just my first exposure to the 3DS' 3D functionality. The original was also extremely impressive looking for the N64 days.
Chrono Trigger - The Trial/Millenial Fair sequence.
Metal Gear Solid - Psycho Mantis, reading the memory card, and codec on the back of the case. Just great stuff.
Half-Life - The intro sequence still sticks with me while I have forgotten particulars of much of the rest of the game. Just amazing at the time.
EVE Online - The first time I flew by some monster-sized ship (during my trial) and realized it was another player's ship.
BBS Door Games - I played my first MUD on a BBS, and things like Trade Wars 2002, Barren Realms Elite, and Legend of the Red Dragon left a huge imprint on my tastes and preferences in gaming.
Thief - Just the entire way the game was played, despite being a "FPS".
Body Harvest - The scale and open world nature of the entire title just amazed me.
Left 4 Dead - The first playthrough of No Mercy with friends, all of us going in completely blind. I was amazed and thought this game would be the greatest thing ever... unfortunately, nothing quite lived up to No Mercy for us.
That's a long and yet still quite incomplete list.