"Proud Of Your Boy". Ultimately cut from the film, but present in some stage productions. The original version is sung by Menken himself; a 2004 Remake was sung by Clay Aiken.
Not only does Mark Mancina's instrumental score for the movie qualify for this trope, but some of those pieces, including "Moves Like an Ape, Looks Like a Man" perfectly fit the Crowning Music of Heartwarming trope.
Think that's chilling, try the song they had planned for Shere Khan that was cut, which showcases his character as a regally ruthless hunter with a cold, dark baritone: it was cut for being too dark, among other reasons as cut characters and characterization.
"Look for the baaare ne-cess-ities, the simple bare necessities...." (Not written by the Sherman Brothers though, but by Terry Gilkyson.)
If you didn't like the simple "Bare Necessities"... then there's always this version of the song by none other than Louis Armstrong. Amazing doesn't even begin to cover it.
On the instrumental side of things, "The Overture" and "Tiger Fight" are pretty much amazing. Tiger Fight from 0:48 to the end is epic and makes driving a very fun adventure.
"OOOOOOOOOH~! Supercalifragilisticexpialidocious/Even though the sound of it something quite atrocious/If you say it loud enough you'll always sound precocious/Supercalifragilisticexpialidocious!"
While James Newton Howard's works with Disney is purely non-musical (save Treasure Planet: the songs were written mainly by Goo Goo Dolls' frontman John Rzeznik) he nonetheless deserves mention for his works in 3 films:
The leitmotif "This Land" which recurs throughout The Lion King—the most poignant occurring when Simba cuddles up against Mufasa's body and later when he speaks to his father's spirit, the most stirring and triumphant when Simba claims the throne in the finale. While the whole movie was incredible, it would not have been half as emotionally effective without this music.
"Circle of Life" surely blows many away, especially when accompanied by the opening animation of the film.
"The King of Pride Rock" takes parts from both of the above, combines them, and adds to them to create a nearly six minute long masterpiece of modern orchestra.
The Broadway version of "King of Pride Rock" takes from several other songs from that version, and ends up with a spine-tingly epic conclusion. (Too bad Simba's Roar of Awesome wasn't included in the soundtrack...) Also of note are "Shadowland", a slow, haunting song, "The Lioness Hunt" and "Grassland Chant", both sung in Zulu, and "Endless Night" which, well, speaks for itself. The whole Broadway production is amazing, actually.
As anyone who's had to play it knows, the finale of Stravinsky's 1919 Firebird Suite is an absolute bitch to play because of the changes in meter. It is also epically gorgeous and heroic, as evidenced in its use in Fantasia 2000.
A short instrumental piece of "Honor to Us All" that plays over the opening titles.
And the music that plays as Mulan slides down the lantern rope near the end of the film, and shortly thereafter the music that plays while the entire population of the Forbidden City bows to her. None of these are on the official soundtrack.
The English version is notable for the fact that it made Donny Osmond, who performs the song, far more awesome than he could ever hope to be on his own.
"Vacuum Rescues the Group", which is just plain epic, with all the fanfare and so on. Also, there was a totally Zelda-style "Item Get" when Toaster gets the battery.
"Worthless", which doubles as a Tear Jerker for the cars being dragged to their final fates in the crusher.
Also, "Toaster's Dream." Yes, it plays during one of the most terrifying scenes in the whole movie, but it is actually very epic and creepy at the same time.
"The Storm". Lampy, the comic relief, nearly getting himself killed just to recharge their battery in a storm to save not only Blanky, but the rest of his friends as well? Coolest. Scene. Ever. And the music just about proves it.
"Why Should I Worry" was one of the most catchy and kickass songs ever to come out from Disney. (Bonus points for being sung by none other than Billy Joel!)
"The Journey," which played over the opening credits, is hauntingly beautiful and sadly overlooked.
Bolt, and in particular "Rescuing Penny", which manages to be sad and awesome at the same time.
Barking at the Moon, in combination with the road-trip montage that it accompanies, is Crowning Music of Heartwarming.
"I Thought I Lost You", proved that after 30+ years, John Travolta still had great pipes.
Peter Pan 2: Return to Neverland had the song "I'll Try". If you don't cry hearing the pain and loss of childhood Jane's suffering having grown up in WW 2, you'll cry when the downtrodden tune gets reused and revamped at the end to be an uplifting song about finding something to believe in despite it all.
The music from The Rescuers Down Under is pretty cool; for example, the music for when Cody is flying on Marahute's back, and the music in the opening.
The Transformation sequence in Brother Bear has a beautiful Inuit song playing over the entire scene which makes this scene even more awesome. The best part, the song is also available in English on the soundtrack and when you know what it actually means it just fits this trope even better.
Special mention goes to Ron Jones' excellent score for the series. Several pieces he composed are suspenseful, melancholic, amusing, exciting, and triumphant; they even rival some theatrical scores. Here's a sample.
The House of Mouse generally has a musical number every few episodes, usually a new spin on a classic Disney tune, including the single greatest rendition of "Who's Afraid Of The Big Bad Wolf?" ever made.
Big Bad Wolf Daddy: "I love to bring down the house!"
SRMTHG: The music played at the very end of the first season finale, and all variations on it. Also, the music played when Antauri first appears as the Silver Monkey, and all variations on that. Hands down.
The intro theme to D-TV (which was a series of music videos using footage from Disney shorts and movies made for Disney Channel back in the day). Does the song scream 1980's? Yeah. Is it still awesome? Yeah.
