Dr. Strangelove
Punch Drunk Love (or any P.T. Anderson movie)
12 Angry Men
How has no one mentioned a single P.T. Anderson movie? He's the Stanley Kubrick of our generation! Punch Drunk Love, Magnolia, There Will Be Blood, Boogie Nights, and Hard Eight; all wonderful movies that deserve people's attention. Go see these movies NOW!
It really is difficult to pick just three movies. I mean I'm leaving out some of my favorite movies like Sidney Lumet's Dog Day Afternoon, recent favorites like Avatar, A Serious Man (should've won best picture), Inception, and even Scott Pilgrim vs. The World (which I saw just last night). There are horror movies like Cannibal Holocaust and Dawn of the Dead. Comedies along the lines of the Evil Dead Trilogy and Animal House. And of course Scorsese: The Departed, Taxi Driver, After Hours, and Raging Bull. We're missing anti-establishment landmarks like Easy Rider, One Flew Over the Cuckoo's Nest, and Cool Hand Luke. Not to mention movies that have disconnected story lines and director styles like Pulp Fiction, Crash, Amores Perros, and Magnolia (P.T. Anderson again.) Cult favorites like the Monty Python movies, Videodrome, The Warriors, Donnie Darko, the recent Drag Me to Hell, and Dead Alive (the goriest movie of all time.)
Foreign films like Metropolis, Life is Beautiful, the Lives of Others, the Red Balloon, Elevator to the Gallows (very underrated), and Inglorious Basterds (that's right I consider a foreign film). Social commentary films like Brazil, To Kill A Mockingbird, A Clockwork Orange, Platoon, and Natural Born Killers.
Classics like Treasure of the Sierra Madre, The Godfather The Dirty Dozen, Citizen Kane, Serpico, Singin' in the Rain, Alien, Night of the Living Dead, and all of the Kubrick movies (The Killing, Paths of Glory, Dr. Strangelove, Barry Lyndon, Eyes Wide Shut, A Clockwork Orange, the Shining, Full Metal Jacket, Lolita, Spartacus, and 2001: A Space Odyssey). Action movies like Bullitt, DIe Hard, First Blood, Speed. Thrillers like The Sixth Sense, 12 Monkeys, Goodfellas, Falling Down, and anything by Alfred Hitchcock (who I haven't seen a lot of in this forum). And don't forget about animated movies from Disney, Pixar, and Studio Gibhli (Fantasia, Up, and Princess Mononoke respectively; also Fantastic Mr. Fox by Wes Anderson).
To keep it simple, it's difficult if not impossible to name just three movies.