Now this may rack some people's brain but what was the first game of each main genre? Some people treasure the first game in a genre they've ever played. My games are:

Shooter: Call of Duty 4 Modern Warfare (PC)
Side Scroller: Disney's Hercules (PSOne)
Platformer: Crash Banditcoot 3: Warped (PSOne)
Action Adventure: Sypro 2: Gateway to Gilmmer (PSOne)
Action RPG: Kingdom Hearts (PS2)
JRPG: Pokemon Ruby (GBA)

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Currently Playing: Guacamelee (PSVita)/ Injustice: Gods Among Us (iOS)
Last Finished: Far Cry 3: Blood Dragon (PS3)/ Platinumed PlayStation All-Stars Battle Royale (PS3 and PSVita)
Last Bought: Far Cry 3: Blood Dragon (PS3)

Currently Watching: Smallville Season 10

Shooter: 007: Nightfire (PS2)
Side Scroller: Sonic (MegaDrive)
Platformer: Crash Banditcoot (PSX)
JRPG: Pokémon Blue (Game Boy)
Racing: Gran Turismo (PSX)

I can't remember the others haha, and these aren't necessarily correct either! They are all games that stuck in my mind in any case.

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Wow, fun topic but terrible for us old gamers with crappy memory (haha). I'll try my hardest!

Shooter: Wolfenstein 3D (DOS)
Side Scroller: Gradius (Arcade)
Platformer: Donkey Kong (Arcade)
Action Adventure: Adventure (Atari)
Action RPG: Legend of Zelda (NES)
JRPG: Dragon Warrior (NES)

Hope I didn't miss anything >_<

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Hmmm, I'm pretty sure these are mine:

Shooter: Dark Forces (PC)
Side Scroller: Super Mario Bros. (NES)
3D Platformer: Super Mario 64 (N64)
RPG: Final Fantasy II (FFIV on SNES)

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Let's see what I can remember...

Platformer: Super Mario Bros.
3D Platformer: Super Mario 64
Racing: Final Lap (Old arcades ftw)
RPG: Pokemon
Sports: Baseball (Intellivision!)

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At the end of it all, gamers play what we play not for "game over," but rather for what we take away.

what? no one here has ever played a puzzle game?

Shooter: Doom (DOS)
Platformer: Pitfall! (Atari 2600) (or Super Mario Bros. (NES) if you can't consider Pitfall in that category)
3D Platformer: Super Mario 64 (N64)
Racing: Rad Racer (NES)
RPG: (depends on whether or not you consider Zelda an RPG) Legend of Zelda (NES) if you do, Final Fantasy (NES) if you don't
Sports: Golf (NES)
Puzzle: Tetris (NES)
Shoot em up: River Raid (Atari 2600 via a collection on PC)
Arcade: Ms. Pac-Man (Atari 2600) (makes sense, seeing as it was my first game EVER
Music/Rhythm: Parappa The Rapper (PS1)
Party/Mini: Chef's Love Shack (N64)

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Oh Wow I need to think about some these Genres...give me a sec.

Racing: Mario Kart 64(N64)
Shooter: Duck Hunt(NES), call me old fashion but that was my first shooter with the orange gun controller if anyone remembers.
Side Scroller: Super Mario Bros(NES)
Platformer: Spyro the Dragon(PS1)
Action Adventure: The Legend of Zelda: Ocarina of Time(N64)
Action RPG: Kingdom Hearts(PS2)
JRPG: Final Fantasy VIII(PS1)
Strategy: Civilization(PC)
Puzzle: Tetris(PC)
Party: Mario Party 1(N64)

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Shooter: Time crisis (i think) (Ps1)
Platformer: Mickey wild adventure (ps1)
Action Adventure: Tomb raider (Ps1)
Action RPG: Kingdom Hearts (PS2)
JRPG: Pokemon blue (GBC)
Racing: Crash Team Racing (Ps1)
Strategy: Worms (Pc)
Puzzle: Tetris(PC)
Party: Buzz (Ps2)
Sports: Fifa World Cup 98 (Pc)

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Top of my head...

Platformer: Sonic The Hedgehog (SMD)
RPG: Pokémon Red (GB)
Puzzle: Tetris (GB)
Racing: Gran Turismo (PS)
FPS: Doom (PC)
Strategy: Worms (SMD)
Adventure: Tomb Raider (PS)

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Shooter: Wolfenstein (PC)
Side Scroller: Double Dragon (NES)
Platformer: Super Mario Bros 1 (NES)
Action Adventure: In the genre's current form I guess it's Tomb Raider 1. (PC)
RPG: Eye of the Beholder (PC)
JRPG: Final Fantasy 7 (PS)
Racing: Wacky Racers (PC)
Strategy: Sim City/Civ 1 (PC) I don't remember which one was the first one.
Fighting: Street Fighter II (SNES)
Sports: Tecmo Cup Soccer (NES)

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<Insert profound comment here>
@Chronometer2300