There's a fuck-ton of games I played as a kid/teen that pretty much defined me and I'd advise anyone to play even now, even if some of the graphics are now outdated.
Oldschool shit:
Sonic 1 (Master System)
Sonic 2/3 & Knuckles (Mega Drive)
Super Mario 3
Turbo Outrun
A little newer but still oldschool shit:
Crash Bandicoot 1-3
Crash Team Racing
Abe's Oddysee/Exoddus
Destruction Derby 2
Gran Turismo 2
Digimon World 1, 2003 (3 outside of Europe)
Tekken 2/3
Metal Gear Solid
Pokemon Yellow, Crystal
GTA 1, 2
Air Combat
A little newer:
Pokemon Sapphire/Emerald
Metal Gear Solid 2/3
Burnout 2,3, Revenge
Metal Arms: Glitch in the System
Super Smash Bros. Melee
Sonic Adventure 2: Battle
GTA 3, Vice City
Ace Combat 04, 5, Zero
Sonic Advance 1, 3
Fire Emblem 7, 8
Last gen:
Metal Gear Solid 4
Super Smash Bros. Brawl
Race Driver GRID
I've played a lot more than what I've listed here, but these are the titles I remember the most/had the most fun with. Maybe there's more that I enjoyed playing, but I can't remember everything right now.