You can't only pick 3. That being said:
Tetris - the basis of most modern puzzle games.
Super Mario 64 - How 3D platforming is done correctly.
Either Super Metroid or Zero Mission - Both are great games that invite exploration and experimentation.
Castlevania: Portrait of Ruin - It takes all of the developments since Symphony of the Night, adds them all up, and improves on them.
Either Gradius V or Ikaruga - Both are Treasure games and top-of-the-line sh'mups.
Legend of Zelda: Link to the Past AND Ocarina of Time - How adventure gaming is done in 2D AND 3D.
Metal Gear Solid - Despite all the later variations, this one is still the best choice for the genre.
Contra - The best side-scrolling shooter. Legendary.
System Shock 2 - The game that made Deus Ex possible. It used RPG, stealth, shooter, and horror elements to maximum effect.
Final Fantasy VII - Yes, I know... we've all heard it before. Regardless, it's the game that got me to start playing RPGs and despite the entry-level existentialism, it started my love of philosophy. So
And, it's very non-RPG-player friendly. It's a good starting point, regardless of criticism.
Tetris - the basis of most modern puzzle games.
Super Mario 64 - How 3D platforming is done correctly.
Either Super Metroid or Zero Mission - Both are great games that invite exploration and experimentation.
Castlevania: Portrait of Ruin - It takes all of the developments since Symphony of the Night, adds them all up, and improves on them.
Either Gradius V or Ikaruga - Both are Treasure games and top-of-the-line sh'mups.
Legend of Zelda: Link to the Past AND Ocarina of Time - How adventure gaming is done in 2D AND 3D.
Metal Gear Solid - Despite all the later variations, this one is still the best choice for the genre.
Contra - The best side-scrolling shooter. Legendary.
System Shock 2 - The game that made Deus Ex possible. It used RPG, stealth, shooter, and horror elements to maximum effect.
Final Fantasy VII - Yes, I know... we've all heard it before. Regardless, it's the game that got me to start playing RPGs and despite the entry-level existentialism, it started my love of philosophy. So