No, not three. Three for an area of focus maybe.
Shooters
Doom - A kill ever damn living thing! fragfest, plus it more or less started the genre
Half-Life 2 - A showcase of story-telling with more thoughtful, slower-paced combat
Halo: Combat Evolved - A mix of both, a benchmark for the console shooter, and a heavy influence in the design of most shooters made since
Computer Role-Playing Games
Final Fantasy VII - A Japanese RPG
Star Wars: Knights of the Old Republic/Mass Effect (student choice) - A Sci-Fi RPG with non-linear mission progression and explorations of morality
The Elder Scrolls III: Morrowind - A fantasy RPG with open world sand-box gameplay
Strategy
StarCraft - The phenominal popularity just begs study and it is highly representative of Real-Time Strategy
Sid Meier's Civilization - Turn-Based Strategy
Rome: Total War - Real-Time Tactics
Nintendo
Super Mario World - The most representative form of Mario
The Legend of Zelda: A Link to the Past - Again, this is the basis of Zelda
Metroid Prime (play-through of included Metroid classic required) - Both models of Nintendo's adventure platformer
Racing
Star Wars Episode I: Pod Racer - Sci-Fi racer
Mario Kart - Kart racer
Project Gotham Racing 4 - Realistic racer
Stealth
Thief: The Dark Project - First-Person Stealth
Metal Gear - Isometric stealth
Tom Clancy's Splinter Cell - Third-Person Stealth