Oof, this question is alway hard since there's been a BIG list of games I've loved for years and still love today despite their age. So, I'll make a list based on the genres I've played.
First Person Shooter: Perfect Dark Zero, TimeSplitters 2, Metroid Prime, Perfect Dark, Star Wars: Republic Commando,
Multiplayer-Only Game: Team Fortress 2, Fistful of Frags, Star Wars Battlefront (the early 2000s duology + handheld spinoffs), Garry's Mod, The Mean Greens: Plastic Warfare, Human Fall Flat
Run 'n' Gun: Cuphead, Centipede: Infestation
Platformer(s): Toy Story 2: Buzz Lightyear to the Rescue! , Super Mario Sunshine, A Bug's Life, Donald Duck Goin' Quackers, Battleblock Theater, Spongebob Squarepants: Battle for Bikini Bottom, The Spongbob Squarepants Movie (2004), Sly Cooper, Rayman, Oddworld: Abe's Odyssey (& Exoddus), Crash Bandicoot (starting from Wrath of Cortex onward AND backward), Yooka-Laylee, Super Mario, Sonic the Hedgehog, The Emperor's New Groove, Shantae: Risky's Revenge, Pac-Man, Fairly OddParents: Breakin' da rules, Jimmy Neutron: Jet Fusion, Ty The Tasmanian Tiger
Single-Player Action Game: Grand Theft Auto 4, Stranglehold, The Incredibles, Half-Life, Tomb Raider (the entire franchise), Uncharted (the entire franchise), Batman: Vengeance, Superman: Shadow of Apokolips, Centipede (1998), Retro City Rampage, Double Dragon Neon (it's also multiplayer, but single is just as fun), Prince of Persia: The Sands of Time
Racing/Sports: Burnout, Need for Speed (prior to the 2015 reboot onward), Gran Turisimo, Disney's Extreme Skate Adventure, Mario Kart, Madden 2000 and 2008, FlatOut, and Crazy Taxi.
I have a lot, and to be honest, I feel like I missed a lot on this list. But I feel like I'll end it here so I don't overload everyone.