For my first entry, I'm gonna try to remember what games I have completed the story mode or its equivalent at least once. I'll try to keep up with this as needs be, and update when needed.

Teenage Mutant Ninja Turtles: Fall of the Foot Clan (GB)
Doom II (PC)
Goldeneye 007 (N64)
Grand Theft Auto III (PS2)
Halo: Combat Evolved (XBOX)
Halo 2 (XBOX)
Kirby's Dream Land (GB)
Pokemon Red (GB)
Metal Gear Solid (PSX)
Metal Gear Solid 2: Sons of Liberty (PS2)
NBA Street Vol. 2 (PS2)
Soul Calibur (DC)
Tekken 3 (PSX)
Tekken Tag Tournament (PS2)
Tekken 4 (PS2)
Tony Hawk's Pro Skater 2 (DC)
Tony Hawk's Pro Skater 3 (PS2)
Tony Hawk's Underground (XBOX)
Unreal Tournament (PC)
Who Wants To Be A Millionaire (PC)
Top Gear Rally (N64)
Super Mario World (SNES)
Mario Kart 64 (N64)
Diddy Kong Racing (N64)
Super Mario 64 (N64)
Super Street Fighter II: The New Challengers (SNES)
Street Fighter II: Turbo (SNES)
Killer Instinct (SNES)
Killer Instinct Gold (N64)
Mortal Kombat (SNES)
Mortal Kombat (GB)
Mortal Kombat (GG)
Mortal Kombat II (SNES)
Super Mario All-Stars (SNES) (Mario 1, 3 & Lost Levels)
Super Smash Bros. (N64)
Star Wars Episode I: Racer (N64)
The Lion King (SNES)
NBA Jam (SNES)
NBA Jam: Tournament Edition (SNES)
Super Mario Land 2 (GB)
Castlevania: Symphony of the Night (PSX)
Star Fox: Assault (GC)
Advent Rising (XBOX)
007: Everything or Nothing (XBOX)
X-Men Legends II: Rise of Apocalypse (XBOX)
Super Smash Bros. Melee (GC)
WWE Smackdown! vs. Raw (PS2)
WWE Smackdown! vs. Raw 2006 (PS2)
Quake III: Arena (PC)
Chrono Trigger (SNES)
Teenage Mutant Ninja Turtles: The Hyperstone Heist (Gen)
The Legend of Zelda (NES)
Red Faction II (XBOX)
Need For Speed Most Wanted (PS2, X360)
Need For Speed Carbon (PS2, X360)
Gears of War (X360)
Perfect Dark Zero (X360)
Crackdown (X360)
Lego Star Wars II: The Original Trilogy (X360)

Tom Clancy's Rainbow Six Vegas (X360)

Guitar Hero II (X360)

Stranglehold (X360)

Halo 3 (X360)

Assassin's Creed (X360)

Tom Clancy's Rainbow Six Vegas 2 (X360)

Guitar Hero III: Legends of Rock (X360)

Burnout Paradise (X360)

Super Smash Bros. Brawl (Wii)

Mario Kart Wii (Wii)

Mario Kart DS (DS)

Rock Band (X360)

Guitar Hero: Aerosmith (X360)

Resistance: Fall of Man (PS3)