I don't like how you excluded Dreamcast, but anyways:
5. Metal Gear Solid 2: Sons of Liberty / Resident Evil 4 (TIE)
Mind****. Yes, a total mind**** it was. But damn was it an enjoyable one. The graphics were just incredible for its time, the voice acting exceptionally well done, the gameplay amazing. All the little touches, from shooting magazines out of the rack, to holding guards hostage, this game felt completely next-gen. Though it was a linear game, the amount of options it gave made it feel less like one. I didn't mind Raiden, sure Snake is a cooler character but playing as Raiden didn't really lessen the experience for me. It just gave another side to the story. Crazy ass story it was, but I understood most of it. And Snake jumping after Ocelot at the end was just one of the greatest moments in gaming ever. It was epic, it was crazy, it was awesome. MGS2 remains one of my most memorable gaming experiences ever.
This screen always wows me. As does RE4 whenever I play it. The graphics are ****ing awesome. The best of this gen. A combination of a great art style, kick ass animation, and amazing tech behind it. But graphics are not the main reason why RE4 rules, no no no, its the gameplay. It just feel so fresh, so fun, so bad ass when you shoot a monk in the leg and then proceed to crush his skull with a suplex. The context sensitive actions are one of the best things about the game, they were used perfectly. Whether its kicking a Ganado in the face, ducking under a tree El Gigante tries to swipe you with, or jumping out of a window, it always feels cool, and its always fun. This is an action game, but it does quite a few things to make it feel unlike one. First is the Merchant. Upgrading your weapons, selling stuff, finding treasures, upgrading Leon's health, buying new weapons, this all goes a long way to give the game more depth and an RPG-like feeling. A great number of boss fights, all of them being awesome, and quite a few different enemies keep the game feeling fresh and exciting. The behind the back camera angle was genius, and the aiming system rocked. The game has a few problems though. It doesn't feel like an RE game, its rarely scary, the music ain't special, and the story is pretty weak. However the game is just so fun and replayable that its easy to ignore these things.
4. Super Smash Bros. Melee
The best multi-player game ever. The best fighting game ever. My most played game ever. Yes, I give it quite a few awards, but it sure as hell deserves it. I bought a GameCube because of this game, in fact I bought a GameCube the day this game came out. I've unlocked everything, all levels, all characters, all trophies, and it took me a loooong time, and I loved it. The amount of depth in the game is barring on extreme. Tons of different modes, multiplayer and single player. 290 tropies to unlock, some quite easy, some extremely hard. The game is a complete and total blast with friends, I've busted up laughing at times just because the game is so fun and hilarious. There are countless legendary moments I've had playing this game, and I'm sure there will be a few more up until Brawl's release. Nearly 4 years after release and I still play the game quite a bit.
3. The Legend of Zelda: The Wind Waker
You see that guy there? His name is Ganondorf, and he is the greatest game character ever. I don't care if you disagree or not, he is my favorite, and part of the reason why I enjoyed Wind Waker so much. They really fleshed out his character, and made him much more than the mindless power hungry villain he was in Ocarina of Time (though I like him like that as well). The story in TWW is by far the best in the series, however, the gameplay is also the best as well. Sure it has some problems, mainly being way too easy, but otherwise its extremely polished. The combat was vastly improved, and the puzzles were done extremely well. I loved sailing, I loved how vast the sea was, I liked how long it took to go from one place to another, it really gave a feeling that you could do anything you wanted. I wish they had made combat on the boat fun though. As for the Triforce hunt, the amount of complaining it recieves is overdone. Yes it was tedious, but it didn't really take that long. Lets not forget the graphics, which are just plain incredible. The most beautiful game of this gen (though I've yet to play Okami and Twilight Princess ain't out yet). One of the few games of this gen that will be looked back on 10 years after being released and it'll still look great. The dungeons, while not many, were all great except for the Stealth Dungeon. Tower of the Gods is my favorite dungeon in the entire series, I loved how it utilized everything you had, including your boat. The puzzles were great and I just loved the feel of it. The bosses, while easy, were all quite fun to beat. The final battle remains my favorite boss battle of all time. Overall, my favorite Zelda game.
2. Metroid Prime / Shenmue II (TIE)
First, I shall talk about Metroid Prime. What an amazing game this is. Its hard to believe that Retro Studios was able to create a game like this on their first try, I'm sure they got plenty of help from Nintendo, but still it always astounds me. Prime 1 is very similar to Echoes. They both share amazing graphics and art style, great boss battles, and are polished to perfection. Meta-Ridley stands as my favorite boss battle from either game, and I generally liked the areas in Prime 1 more (excluding Sanctuary Fortress), I loved the Ice Beam, and this game felt a lot more fresh than Echoes did. However I do have a few small complaints, the Plasma Beam was overly powerful, once you got it there was pretty much no reason to use the other weapons except for opening doors, fighting Fusion Metroids, or the first part of the final boss battle.
Perhaps the most epic experience I've ever had in gaming. This was better than the original in every way, and though the cliffhanger ending pissed me off to no end, I loved the game through and through. The best story in a game bar none, and it fixed the little annoying things in the original like having to wait around until a certain time passed. The jobs were more fun, there were more things to do, the cast was better (Ren FTW), more action, more fighting, and just overall was plain awesome. Only problems I had was the dub (while improved from the original) was still pretty meh, and carrying books around sucked. Otherwise, I loved it.
1. Metroid Prime 2: Echoes
Yes, most people seem to think this was a disappointing sequel, but not me. I loved it, it surpassed the first in almost everyway, and was gameplay perfection. Difficulty was set just right, graphics and art style utterly amazing, and some of the best boss battles of this gen. Some people complained about the Dark World, or other parts of the game, but for me I have pretty much no qualms with the gameplay at all. Its perfect, I'm in awe of how polished the game feels everytime I play it. The ammo system added some strategy to combat, and I loved the missile combos (though they weren't as useful as in Prime 1). Sanctuary Fortress is the greatest level in gaming, ever. The only problem I have is the key hunt at the end, however exploring the world with the Light Suit gave you a feeling of how far you have come ability wise. Multi-player was meh, but single player is just pure GREATNESS.
Honorable Mentions:
Grand Theft Auto: Vice City, Marvel VS. Capcom 2, Tales of Symphonia, Donkey Kong: Jungle Beat, Skies of Arcadia Legends.