So many to choose from, a list ensues...
Super Mario Bros (NES), The first game I ever played. Even now on the 3DS hearing that music takes me back to being 5 years old and in awe of someone's ability to control what happened on a TV.
Sonic 2 (Megadrive) The first game I ever got hyped about, have fond memories of poring over magazines with my best friend for hours, arguing over who would play as Tails when one of us got it
Zelda- A Link To The Past (SNES) for every reason stated prior to this post
Zelda- Majora's Mask (N64) the first Zelda game that truly gripped me from start to finish
Pokemon Red (Gameboy)- Spent all of my time when other 16-18 year olds were out getting laid playing through this and blue, transfering Pokemon from one to the other in a bid to catch 'em all, and drunkenly sitting in a local park at 1am squinting at a non- backlit gameboy battling my- equally unattractive- friends
The World Ends With you (NDS) - Just amazing, hard to get into but worth it when you do.
Number One has to go to one of the first 3 Monkey Islands. All 3 are contenders but I think 'curse of' wins out (though the high quality of the puzzles in 2 could swing it)