1)
R-Type for Game Boy was the first ever game I played for myself. Up until then, I'd just watch my brother and sister play games (sister on Game Boy, brother on PC). The R-Type series in general really shaped me as I grew up. Delta was probably the most influential, even though I was in high school by the time that came around and I was out of my formative years, but if it weren't for the Game Boy one, I wouldn't have seen it in the store and went "Whoa, that boss from that game I played a lot as a kid! He looks AWESOME now!!" R-Types came out at around the same time, so I also got to scratch the itch of playing the full version rather than the truncated/simplified GB one, and I NEVER beat them because I am actually TERRIBLE at R-Type games.
2) Honorable mention goes to
Heiankyo Alien (also for Game Boy), which I also played a lot of, and influenced my worldbuilding a lot as I chipped away, for years, at a legitimate explanation for why an alien race capable of space travel could be tricked by a guy with a shovel. I spent so much time worldbuilding for them that they have an honorable corner of my headworld despite not actually being mine. Also, for being one of my most-fanarted games in the shadows, when most people don't even know it exists (it's very obscure unless you're into Japanese gaming historica).
2.5) Also honorable mention goes to
Mega Man X, for being the game with the fandom that spurred me into taking art seriously.
3) Although I grew up loving game music, I think the first one I ever actually loved so much that I would just turn the game on to listen to it was
the opening to "Shanghai" for Game Boy. You know, that old Mahjong Solitaire game? Yeah. I dunno, I loved its title screen song as a kid.
It's such an obscure song that it's almost impossible to find an example of it online, the only video I can find that shows it off is this one. It's a VERY short loop, the video cuts barely a second off the end before starting the game, but yeah. That song. I could listen to it for ages when I was young.
I MADE MY OWN VIDEO OF IT somehow I had a recording of the song lying around. From 2004. I don't even know how but there it is.
It's really hard to pick out any over the others because of how much game music I adored. Anything from Mega Man X/2 to Super Mario RPG to Tetris Attack. Oh man, Tetris Attack. Gosh, I really could go on forever, couldn't I?
4) I make a lot of references to the Game Boy forming the basis of my gaming life, but I never actually owned one (the original one, at least). I mostly watched my sister play hers, and most of the games weren't even hers, they belonged to her friends/boyfriends. I eventually 'inherited' most of those cartridges (she never returned them and eventually handed them all off to me when she stopped playing games). I spent most of my time playing on other people's consoles (which meant a lot of Sonic, actually, since most of my friends had Geneses), either my friends' or cousins', until I finally got a SNES of my own, near the end of the SNES' lifespan, and since my parents rarely bought me games, a lot of it subsisted on Blockbuster rentals. Most of my early gaming prior and parallel to that, Game Boy aside, was on my dad's PC, with games like Lemmings (ALSO a big formative one), Math Blaster, Math Rescue, Super Munchers, SimAnt, Prince of Persia, Raptor: Call of the Shadows, and Mechwarrior/II.