Even though it was just last year, Super Mario Galaxy would probably be it for me. No other game in recent memory threw that much variety and fun at the player all the way through to the very end... and then tossed in a great bonus for your efforts, too.
Older stuff that just captivated or blew me away that I wish I could play again for the first time:
Space Harrier - Although I loved games as a kid like Donkey Kong, Pac-Man, Jungle King, Zaxxon, Dragon's Lair, Q*Bert, etc., it was this game from Sega that made me fall in love with sprite scaling. I'd never seen anything like it. The colors, the robotic and flora/fauna designs, and the great music just won me over. When I saw the sit-down cabinet at Disneyland circa 1987, I think I spent more time in the Starcade than I did with my family waiting in lines for rides. This was *it* for me, and playing it today really takes me back.
Out Run - Pretty much the same as above, but stepping into the driver's seat of a Ferrari as a kid in junior high made me feel pretty damn cool. LOL I honestly thought I could feel wind in my face while playing it. And then the first time I heard Magical Sound Shower, I knew that'd be a favorite for life.
Beatmania IIDX 3rd Style - I'd never seen a machine like this in arcades. Up until they started importing this to some US arcades, all I'd seen was DDR and Beatmania Complete Mix 1 & 2. This machine was like, hooooly crap. The cool (at the time) RP screen, huge speakers, platform you had to stand on, LED readout, FX sliders, the 14-key setup, real videos playing. Man... those were the days.
Ico - All about atmosphere, and man, did they nail a great sense of scale and lighting. I showed this off to everyone I knew, but few "got it". Some folks couldn't get past the low resolution, some thought it was boring. I found it to be one of the most original and rewarding experiences on the PS2. Minus that horrid US box art, of course.
Chrono Trigger - The sky shot, the balloons, Crono waking up, and the gentle music that plays. So good. This game did so many things that gave RPGs a refreshing feel, and the court scene was one of the coolest things I'd ever seen in a game up to that point. The talent involved in this game was also unheard of, and getting all the cool pack-in posters and the nice, thick manual made this a great deal.
Final Fantasy V - The music. Damn, it was cool hearing music this good coming from the SFC. And then the credit sequence, and hearing Ahead on our Way was just... wow. I hadn't even taken Japanese in college yet, but games like this made me want to learn that day.
Yoshi's Island - The intro was whatevers (cute music box, but I didn't like the visuals), but upon starting the first level, I was unprepared for how this game looked like in motion. Screenshots were cool, but the subtle touches (like the flapping butterflies in the foreground) really made an impression on me.
Thunder Force IV - Just seeing/hearing that intro for the first time was incredible. And then when the first stage loads, OMG, parallax insanity.
R-Type - Art style was so new, clean, and polished. First time you play that 3rd mothership stage was a really novel concept (and a frustrating one if you died near the end of it... back to the beginning with you!).
The list goes on and on. It's basically a "your favorite games" thread, but I tried to just talk about how they made an impression on me from the first moment I played them.