The winner for the most unexpected piece of music to ever come out of the Kingdom Hearts series: "A Date With Fate", a bouncy remix of "Succession (Theme of Crisis Core)" that plays during scenes with Zack.
The Unversed Boss music, particularly "Unforgettable", is fun, as well as giving huge hints that the Unversed come from Vanitas due to the similarity of the themes.
"Dismiss", the Final Boss theme, takes the already awesome "Fragments of Sorrow", mixes it with Aqua and Terra's themes, and turns the whole thing Up to Eleven, bringing out both a feeling of foreshadowing and tragedy as Aqua has to defeat her possessed former friend.
"Makaukau", the Deep Space battle music. It's pretty much hardcore techno.
These two remixes of "Scherzo Di Notte" and "Radiant Garden", which take the already great original songs and adds a touch of grandiosity fitting for Kingdom Hearts: Birth by Sleep's version of the world. And it's even more awesome when you realize this is a song of Radiant Garden a.k.a. Hollow Bastion in its glory before Xehanort ruined it and Maleficent took it over.
"The Tumbling", otherwise known as the Wheel Master battle. Fans were rearranging and remixing this track almost immediately after it was revealed in an early gameplay trailer.
"Rage Awakened -The Origin-". For all the awesomeness from KHII, it's even better here. It has a sense of urgency that fits the tone of the fight that KHII didn't have. The song makes that scene.
"Terra's Theme" isn't bad either and befit Terra's resolve and will. It is also similar to "Rage Awakened" since both songs are affiliated with Terra, either as himself or as the Lingering Sentiment/Will.
Considering that Sleeping Beauty had a soundtrack entirely comprising classical music, it's no surprise that the world based on it has fittingly great music. Having said that, listen to "The Rustling Forest", the battle theme of Enchanted Dominion, and try not to go deaf from how awesome it is.
Impossible as it may seem, the Ice Cream Beat mini-game actually managed to come up with a version of "It's A Small World" that's not only not-cringe-inducing, but actually pretty good! (it helps that the thing that makes the original so ear-grating—the singing—is absent).
From the game's Final Mix we have "Master, Tell Me The Truth", the battle theme for The Armor of The Master, Master Eraqus' Sentiment, which incorporates some of the Land of Departure's battle theme.
Also from the Final Mix, "Forze dell'Oscurita", which is the battle theme of No Heart, Master Xehanort's Sentiment. It epically combines "Forze del Male" with "Darkness of the Unknown" for music of sheer power. There's some Fridge Brilliance with "Forze dell'Oscurita", as well: not only does it contain part of Xemnas' final battle music, but the section it borrows comes from when you're fighting Armored Xemnas, whose appearance is almost identical to No Heart's.
The version of "The Encounter" used in Birth By Sleep may be superior to the original! And it gets used in an awesome way in 3D when it plays during the JuliusBonus Boss fight.
The haunting "Sunset Horizons" that plays in the Keyblade Graveyard really gets across the tragedy of what has and will transpire in that place. It doesn't matter that you just spent 30-40ish hours playing as these characters. THEY. WILL. LOSE. Interestingly, it has some of Aqua's Theme laced into it, hinting that she is the only one to truly survive the encounter with Xehanort and Vanitas here.
"Eternal Moments", an extraordinary remake of 358/2's "At Dusk, I Will Think of You" that plays when Ventus gets to first meet Lea and Isa in Radiant Garden. Also, note that the music has more slightly upbeat feeling compared to "At Dusk, I Will Think of You" which sounds relatively calm and slow. This is to indicate how the characters still have heart, as opposed to them as nobodies without heart in 358/2(at the time, at least).
"Tears of the Light", plays at Terra's first scene in Castle of Dreams and Aqua's first scene in Dwarf Woodlands.