Wow, I never noticed this topic. There's plenty to choose from for me, but it wasn’t difficult to come up with a list of just five.
In no particular order:
Mega Man X (SNES) - Love this game; it was the first one (alongside Operation C) that I bought with my own money.
Pokémon Red Version (GB) - The main culprit of my obsession with Pokémon.
Final Fantasy VII (PS1) - The first game I owned on the PS1, as well as the first RPG I played knowing what the genre was.
Teenage Mutant Ninja Turtles: Turtles in Time (Arcade) - I spent a good portion of my childhood playing this game.
The Legend of Zelda: Link's Awakening (GB) - I would spend countless hours playing this game, especially at school during recess. I had a friend who had already beaten it, so he would usually help me out, especially when finding heart pieces.
Obligatory honorable mentions:
StarFox (SNES)
Super Mario Land: 6 Golden Coins (GB)
X-Men vs. Street Fighter (Arcade)
The Simpsons (Arcade)
Pokémon Gold Version (GBC)
I have great memories playing these games.