I've tried to limit these games to those which you can't add your own music to, and to those that have, for the most part, original music. That means the Kirby puzzle games won't qualify and neither will Audiosurf. Now then. Let's get this started!
5. Meteos (by Seiji Momoi, produced by Tetsuya Mizuguchi)
If you're familiar with Rez, you'll understand why the soundtrack of Meteos is so unique; it's by the same producer after all. Rather than being the sort of soundtrack you'd sit down and listen to by itself, the soundtrack in Meteos is dynamically formed as you play the game. Each planet in the game not only has a different appearance, distribution of tiles, and physics, but the music is, of course, different as well. The music is typically just a fairly simple loop however every action you take and match you make plays a sound effect that blends with the music. This means that not only does the music get cooler the better you do, but it also seamlessly gets more frantic as the gameplay does. It might not be the most exciting to listen to by itself, but in-game it's quite a satisfying effect.
Lumines has a soundtrack by... wait a minute- it's produced by Tetsuya Mizuguchi again?! What is this madness? Well let me answer my own question by saying the audio in Lumines changes dynamically as you play, much like Meteos and Rez. Just like in Meteos, there are a variety of "skins" that change the graphics and background music, and also like Meteos, whenever you make a match with the tiles additional sounds and tones will play which blend right in with the music. The better you do, the more complicated and awesome they get. If you get a really good combo going, you'll be rewarded with cooler music to jam to as long as you can keep it up. Once again, the best way to experience this is to play it yourself or see it in action, but I'd say it does make better listening music by itself than the soundtrack of Meteos.
3. Tetris series (by Hirokazu Tanaka and others)
If you're familiar with any puzzle game music, it's likely to be the Tetris themes from the GameBoy version. If you ever questioned whether the famous "Type A" music was originally composed for the game, your suspicions would be founded as it is actually an instrumental version of a Russian folk song. The "Type B" theme is nearly as iconic and was likely composed by Hirokazu Tanaka, who also composed for classic NES games including Metroid, Balloon Fight, Kid Icarus, and Dr. Mario as well as for EarthBound.
But that isn't the only good Tetris music out there. Tetris Battle Gaiden, which was released only in Japan, has music that goes along with its battle-themed gameplay. Here's a playlist of that soundtrack, of which I personally recommend "Ninja" and "Wolfman". Then there was Tetris Attack, the English localized version of Panel de Pon, which had pretty mellow music for the most part. From the full playlist, my personal favorite is "Lip's Theme", which I'm familiar with from its newly arranged version in Super Smash Bros. Brawl.
I'm sure there is plenty of other music from the Tetris series as well, but in order to prevent this post from focusing entirely on Tetris music, I'll finish up and draw your attention lastly to Tetrisphere. Differing from other Tetris soundtracks, Tetrisphere sports a pretty awesome electronic/techno soundtrack. Even if 90s techno isn't quite your thing, this soundtrack still is worth a listen.
When I first played Intelligent Qube, I laughed. I laughed partly because I was playing the Japanese version and my friend and I were struggling to figure out how to play, but the main reason I laughed was because this game's music is far more epic than the game itself is. The game is a fairly unique puzzle game where you control a little guy and have to destroy certain blocks while not destroying others and all the while making sure you don't get crushed. It was an early Playstation game so there isn't much more to it than that. Apparently the memo the composer got was "Write some music for a new Star Wars movie" because what this game got was epic, triumphant, movie quality music. Maybe there is no greater purpose to solving these puzzles, but the soundtrack would have you believe there is. Even if the fun and original gameplay isn't enough to keep you playing, you'll find that it's a joy to play just for the music. This soundtrack truly deserves your attention.
1. Mr. Driller: Drill Land (by Masaru (Go) Shiina)
Masaru Shiina, also known as Go Shiina, is one of my favorite game music composers and only misses out on being my definite favorite because I can't commit to choosing one. I don't own a whole lot of physical CDs, but two of my favorites that I do own are soundtracks by Masaru Shiina. I received the Tales of Legendia soundtrack as a gift after having listened to it online and before I even owned a PS2 on which to play the game it was from. My eventual purchase of the game was much influenced by my love of the soundtrack. The first time I studied abroad in Japan I also picked up Shiina's soundtrack for God Eater, which I also haven't played. So with that said, let me assure you that not having played this Japan-exclusive puzzler for GameCube will not in the least diminish your enjoyment of the soundtrack, which in my opinion, is some of Shiina's best. His unique, but varied, and often jazzy style of music takes full advantage of the varied locations in Drill Land and the many instruments at his disposal. If you can't listen to the whole soundtrack, my personal recommendations are "The Dragon's Feed is the People", "Heading Back to Town", and "Swordray", but really, it's all great. And go ahead and check out the rest of his music while you're at it!