Starting from 3rd Gen since that's as far back as my experience goes...
3rd Gen - NES/SMS -
Super Mario 3: Flying around was so fun xD
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Maniac Mansion: The point and click style was super unique at the time I played it (didn't play many on the PC until later in life)
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River City Ransom: I loved how it had soft elements of an RPG while being a side scrolling beat-em-up
4th Gen - Genesis/SNES and Game Gear/Game Boy -
Sonic The Hedgehog: I sucked at it, but it was a staple in video game history.
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Pokemon Blue: A classic
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Mortal Kombat: Seeing blood for the first time was almost taboo, I remember. xD Lots of buttons were mashed!
5th Gen - PS1/N64/Saturn -
Super Smash Bros: Had an easier learning curve than Mortal Kombat, plus I was insta-cool when I brought my copy to summer camp. xD
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The Legend Of Zelda: Ocarina Of Time: I can always replay this game without getting sick of it. <3
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Goldeneye 007: One of the most influential FPS games!
6th Gen - Dreamcast/PS2/Gamecube/Xbox and PSP/DS -
Grand Theft Auto: San Andreas: Best 3D GTA in my humble opinion. (GTA V is a close second, followed by GTA 2.)
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Super Monkey Ball: The first Gamecube game I played, and still haven't found another series like it.
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Animal Crossing: You have no idea how important this game was to a girl that had no friends, lol. :,)
7th Gen - PS3/Wii/360 and PSV/3DS -
Beyond Two Souls: The first game I played of that cinematic, Quantic Dream-style of games that I absolutely fell in love with.
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Guitar Hero 3: This game was the gateway into a genre that I would sink so much of my life into xD Yay, plastic guitars!
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Wii Sports: A great party game for literally anyone, and was great at showcasing the motion controls.
8th Gen - PS4/X1/Wii-U -
Marvel's Spider-Man: An amazing open world game that looks absolutely amazing.
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Monster Hunter World: Slaying big baddies with friends for the first time was super fun!
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Wii Party U: A fun party game that utilized the game pad really well!
9th Gen - PS5/XSeX/Switch - Probably going to sit this one out until more games come out for this gen.