Halo: Combat Evolved
Outside of DOOM and Goldeneye, Halo is the most successful and prolific old shooter of all time. Halo also introduced multiplayer mechanics to the first-person-shooter world on a console, which completely melted the current PC-only competitive gaming world. Without Halo 1, every competitive first-person-shooter would be different.
Final Fantasy VII
This is widely regarded as the best Final Fantasy game (FFX is a contender, it just depends on what you like). This was the first commercialized JRPG that made its way to a 3D console successfully. Also, playing as Cloud Strife was super cool, the story was memorable, and this game still holds up to this day.