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"The Ancient Temple", a great way to start your first forays into the mysterious labyrinth to perform the trials of the light.
"Searching for Princess Claire", the dungeon theme heard in the later half of the game as you search the labyrinth proper for the hero's father and the princess, it is quite epic.
"Fateful Battle", if you have ever beaten Shining Force 1, you'll immediately recognize this theme, which originated from this game. Only played at two points in the game, once when facing the Dark Knight and again when battling the final boss, naturally. And it is worth it!
Any soundtrack for a Sega Genesis game composed by Howard Drossin, most of which pushed the sound processor to its limits. Try Sonic Spinball, Comix Zone, and The Ooze.
The 16-bit Phantasy Star games have plenty of awesome music to speak of:
Every single track in Phantasy Star II is awesome, but special mention must be made of the opening title sequence, Death Place (which is used as boss music), Mystery, and perhaps the most awesome of this game's awesome music would be this piece, called Rise or Fall. Final boss music? No, that's the basic battle music in PSII!
Phantasy Star III picked up the gauntlet thrown down by its predecessor in the awesome music department.
The music for the penultimate dungeon, Lashute, is sheer, unadulterated horror, with its pounding two-note bassline and viciously sinister organ melody. And it's awesome.
The same could be said for "Maia's Abduction", which kicked off the entire adventure in the first place. Never let it be said that 16-bit sound hardware is incapable of conveying a character's bubbling, seething rage and hatred born of age-old wars.
Searren ~Type 386~, which plays when Wren Transforms into the Global Airship or Aqua skimmer, is as energetic and adventurous as a tune that plays when you have the freedom to fly over the world or sail across the water should be.
Laughter. Truly a shame that it was only used twice in the entire game - when you fight Zio and later the reincarnated Lashiec.
Requiem for Lutz was sad and haunting and conveyed the feeling of thousands of years of history and melancholy in the Espers' Mansion to a tee. It's also interestingly reminiscent of the Phantasy Star IIImain theme.
This piece, which tends to play as the theme of female party members who have a connection to the Algo system's past has a similar feel, invoking feelings of sadness for the lost history of a bygone age.
The Age of Fables, another track that was sadly only used twice, will give you goosebumps. Especially if you hear it in its in-game context - that of the truth being revealed, the final piece of the puzzle falling into place and completing the series' epic story at last.
Organic Beat deserves special mention, both for being objectively awesome, and also for how it is used in the story. This is the music that plays the first time you enter the fort of the game's first Big Bad, which is the moment in the game you realize you're up against something far older and more frightening than you at this point are prepared for. Pretty soon, shit gets real, and this is the music that lead up to it.
Maruyama Hustle. The opening to that song was what made the Backstage and Flying battles against Trouble Bruin so entertaining. You are grabbed by a metal claw rigged to a rocket, dragged through the background, and Maruyama pulls you at extremely high speed through the backstage trying to smash you into scaffolding.
Ohnami-Konami, for the slightly unnerving Puppeteer, and I Sing, which marks the point where the game starts to get really weird.
Made all the more awesome by Planet Sonata making the BGM into a gameplay element. Round 4-1 is one big MacGuffin Escort Mission where you must literally build the BGM a piece at a time by transporting metronomes through the level to a composer. At the end of the level the composer summons a mini-boss that not only attacks in time with the now-complete tune, but times its attack phases to the different phrases in the song.
Earthworm Jim. Who could forget this lovable worm and his game full of music goodness? If the music were chocolate then we'd be fat.
What the Heck? especially. It may not be strictly the best of the game's OST, but it's certainly the most entertaining.
Andy Asteroids?. Who would've thought one of the most awesome tracks in the game would involve a banjo?
While Lufia II had many great tracks, the Boss Theme stood out as one of the most orchestrated-sounding pieces of the SNES's early days, proving that a lot could indeed be done with mere synthesized music.
Speaking of Lufia II's music, how about The Land Nobody Knew? It's incredibly fitting for an isolated town that happen to be situated right underneath Doom Island.
The intro music to Spider-Man/X-Men: Arcade's Revenge is unbelievably awesome, sounding like something straight outta Guns 'n' Roses. It really gets you pumped to see what's in store (though you may feel betrayed once you see how Nintendo Hard the game is).
Also, many of the songs in 1994's Aladdin video game. They sound best in the PC version.
Romancing SaGa 3 is an Awesome Soundtrack. From the Light King's Palace to the battle theme to the Winter Galaxy. It also pushes the SNES's sound chip to its limits with orchestral sounds.
As The Angry Video Game Nerd points out, the Terminator Sega CD version's music. According to his IMDB Biography page, with this game, Tommy Tallarico became the first person to ever use a live guitar and 3-D audio in a video game. Awesome and innovative. You can't beat that.
Soul Blazer is FULL of truly good music... especially this one.
The music playing for the first track in any national cup, or just the Stonehenge track from Top Gear. Once those opposing cascades kick in, you'll feel the hair on your neck stand up.
Also worth checking out: the music from the inevitable Mine Cart Madness level. Starts off as a dark classical tune, then changes keys and sounds like something from a carnival, before finally turning into a jazzy syncopated number in the end. The level may be frustrating, but the music more than makes up for it.
Bio-Hazard Battle isn't exactly a household name, but it had an epic soundtrack, especially for a Genesis title. The background track from Stage 2 and 6 is particularly memorable.
Though Bonk's Adventure (a.k.a. BC Kid) was perhaps best known on the TurboGrafx 16, there was a sequel, Super Bonk, released on the SNES. That game's outer space levels had this epically awesome theme.
An after-market example - Eternal Champions has some pretty good music, but this mash-up of the game's extremely gory finishers is great. To sum up; the game is about a god of balance bringing together several fighters from various points in history who are murdered before their time, and before they would've changed the world for the better. The winner of a to-the-death contest receives the foreknowledge to avoid their death. While the finishers make Mortal Kombat look tame, setting a compilation of them all to Masterplan's "Spirit Never Die" somehow makes the struggle of each character and their quest to win their lives much more emotional.
The opening theme for Maximum Carnage is freaking awesome in fact the whole damn soundtrack for the game probably counts. Green Jell˙'s goal may have been to make music So Bad, It's Good, but with Maximum Carnage, they went for headbangingly awesome metal.
Anything from Power Instinct: Matrimelee. Here are a couple of tracks:
To add more to that, all of the songs are about something miniscule or off-subject. Take for instance, Clara and Anny's song Small Happiness, which is about a depressed girl getting even more depressed about the waitress at the restaurant she's eating at taking her most satisfying thing ("Small Happiness") from her: POTATO WEDGES!
Rise of the Robots was a game whose soundtrack was promoted with Blatant Lies— Brian May's name was mentioned prominently on the packaging, but he only contributed one very short tune for the game. The rest of the soundtrack wasn't even rock; it was techno by Richard Joseph. Yet interestingly enough, Joseph's tunes are actually quite good in their own right, and hold up quite well 16 years later. Check out Loader, Builder, and Supervisor.
Mr. Nutz was a cutesy platformer from 1994 featuring a mascot character who, in stark contrast to most of the video game mascots of the '90s, barely had any attitude at all. And yet, its music is quite amazing. Just take a listen to Woody Land Stage 3, an atmospheric tune that goes for a whole three minutes without looping— and then, once you've done that, listen to this hard-rock cover (uploaded by the original composer!), which somehow makes it even more awesome. Adventure Park is also quite catchy.
Arcana features Second Armageddon, the music for the game's final battle. Jun Ishikawa does a wonderful job of emphasizing the all-or-nothing nature of the fight, especially since by this time Rooks stands alone against Empress Rimsalia and he's the last hope in the darkness, the only thing that stands between her and the total destruction of Elemen.
Super R-Type. Sure the game was Nintendo Hard but it opened with tracks like Solo Sortie and just kept going. The worst part is, the song was long enough for two stages, and the second half can only be heard from the main menu.
Obscure though it is, Namco's 1989 arcade/Megadrive shmup Dangerous Seed is often lauded for its soundtrack. Some of you may recognize the Megadrive version of the themes for the Strike-Ants and Roller-Snail battles, considering they were drafted for Sonic Megamix.
Hudson's Super Star Soldier, and its sequel, Soldier Blade, both on the Turbo-Grafx 16.
Yume Wa Owaranai, the Tales of Phantasia opening theme. Not only because it was a very epic J-Pop song, but it was the first (and possibly, only) song on the SNES to be fully voiced, something developers of that time thought was impossible to do without a CD-ROM add-on and costly co-processor chips.
And last but not least, the title theme and the music for the final bosses which are, not surprisingly, the Power Rangers theme song, the former including samples of the lyrics, the latter being an instrumental version.
The soundtracks of the Power Rangers Movie tie-in game (both of them!) were also pretty wonderful. The Genesis version uses songs from the TV show (Fight, and 5-4-1 for example), while the SNES version uses original arrangements that have a hard-rocking Mega Man X vibe (Aircraft Carrier, Ivan Ooze).
Uniracers, a fairly obscure game from the people who eventually made Grand Theft Auto (and would also get complaints from Pixar for their use of sentient unicycles), cranked this little gem out of the SNES sound processor...and that is hot.