God, now I'm going to feel bad because I can't possibly remember every game I had or played back then. >.< I'll include everything before I got my Playstation. Any consoles I owned will be bolded.
NES:
Teenage Mutant Ninja Turtles
Super Mario Bros./Duck Hunt
Super Mario Bros. 2
Super Mario Bros. 3
Dr. Mario
Tetris
Karate Champ
Wrath of the Black Manta
Batman
Bad Street Brawler
Win, Lose or Draw
Legend of Zelda
Double Dragon
Ninja Gaiden
Mickey Mousecapade
Techmo World Wrestling
Kid Icarus
Punchout
Astynax
Friday the 13th
Ghouls N' Ghosts
A couple of the Megaman games here and there, not sure which though.
SNES:
Super Mario World
Super Mario World 2: Yoshi's Island
Mario Kart
Mario Paint
Zombies Ate My Neighbors
James Bond Jr.
The Legend of Zelda: A Link to the Past
Math Blaster
Super Scope 6
Yoshi's Safari
Street Fighter 2
Mortal Kombat
Mortal Kombat 2
Michael Jordan: Chaos in the Windy City
Super Mario World 2: Yoshi's Island
Mario's Time Machine
Killer Instinct
Tiny Toon Adventures: Buster Busts Loose
Speedy Gonzales Y Los Gatos Bandidos
Bugs Bunny in Rabbit Rampage
Ys III
Donkey Kong Country
Donkey Kong Country 2
Donkey Kong Country 3
Final Fight
Congo's Caper
Super Bases Loaded 2
Wrestlemania
Techmo Super Bowl
NCAA Basketball
Spiderman and the X-Men
Batman Forever
World Heroes 2
Alien Versus Predator
Off-Road Extreme Baja ("Arriba!")
Super Punchout
Teenage Mutant Ninja Turtles: Tournament Fighters
Indiana Jones Trilogy
Star Fox
Rocko's Modern Life: Spunky's Dangerous Day
Game Boy:
Super Mario Land
Super Mario Land 2
Krusty's Funhouse (that Itchy and Scratchy game, I believe)
Legend of Zelda: Link's Awakening
Pokemon Red
Mortal Kombat 3 (*shudder*)
F1 Grand Prix
Home Alone 2
The Lion King
Computer:
Tetris
Word Munchers
Number Munchers
Oregon Trail
Myst
Barney Carnage

(yes, the beautiful Mac game that let you kill invading Barneys with knives, guns and nukes)
Home Alone
Game Gear:
Shinobi 2
Sonic the Hedgehog 2
Sega Genesis:
Sonic the Hedgehog
Sonic the Hedgehog 2
Sonic the Hedgehog 3
Sonic and Knuckles
Sonic Spinball
Robotnik's Mean Bean Machine
Mortal Kombat
Bubsy
Chuck Rock 2
Chakan
Streets of Rage
Streets of Rage 2
Streets of Rage 3
Chase Headquarters 2
Pacman
Tiny Toon Adventures: Buster's Hidden Treasure
Light Crusader
Shinobi III
Michael Jackson's Moonwalker
Tazmania
Jordan vs. Bird
NBA Jam
Toe-Jam and Earl
Kid Chameleon
X-Men
Comic Zone
Desert Strike
Stargate
Fatal Fury
Cool Spot
Terminator 2
Evander Holyfield's Real Deal Boxing
Golden Axe
Golden Axe II
Spiderman and the Sinister Six
Bram Stoker's Dracula
Shaq Fu
Toki
Nintendo 64:*
Diddy Kong Racing
Star Fox 64
Super Smash Bros.
Goldeneye
Dreamcast:*
X-Men: Children of the Atom
*After the Genesis, I would not see another new console until the Playstation. The few times I got acquainted with the N64 were when I visited my cousin in New Jersey, when my sister was sick in Children's Hospital, or during the last few days of grade school during my middle school years. I
only ever got to play the Dreamcast during my school's yearly May Fair. So, I missed a huge portion of games in that time. First things I got to play when PS1 was out were Crash Bandicoot and Tekken 2. As time went on, Resident Evil 1 and 2, Metal Gear Solid, Jet Moto, Dynasty Warriors (the original fighting game), Tomb Raiders 1-3, Mortal Kombat Trilogy, Final Fantasy 7 (yes, I was introduced to the series through VII), followed by 8 (bleh) and then 9 (awesomeness), and somewhere in between all of that was Legend of Dragoon (yay Lavitz! meh everything else), Legend of Legaia, Chrono Cross, Syphon Filter, Lunar 2: Eternal Blue, backtracked to Lunar: Silver Star Story, some PS1 Demo discs (which were usually awesome), Bushido Blade, the awesomeness of the Suikoden series, Castlevania: Symphony of the Night, Ace Combat 6, Rayman, Warcraft 2, Final Fantasy Tactics and so on and so forth. The future was filled with PS2, catching up on old games via Emulator, playing computer games like Doom and Descent, and so on.