This is going to be a hard one, there are lots of really cool games, both named and unnamed (though most named here good choises everyone, some of them I had already fogotten.)
1. Super Smash Bros Melee: It's the best multiplayer game ever. No-one can go against the power of Ninty's beat-em-up. Not even that wannabe Shrek Superslam. There is just something to the simplicity of the game. People pick up a controller play SSBM and have fun you dont need much explanation and soon the button mashing (wich doesnt work) turns into real attacks. Great game and my nr 1.
2. The Legend of Zelda Majora's Mask: Good Graphics even today and a great story.
3. Chrono Trigger: One of the most enticing games I've ever played. I love the FF games and would have placed FF 7 on 2nd if it wasnt for the fact that it's sony's console or the PC you have to play it on.
4. The Legend of Zelda Ocrina of time: What can I say about an almost perfect game?
5. The Legend of Zelda A link to the past: To me still the best zelda story, It was long and captivated. OOT and MM were great too but the story wasnt as good (advantage to MM and OOT is that the Graphics and 3d Worlds sucked you in there)