The "I Want" song (also called an "I wish" song) is a popular type of song featured in musical theatre, and has become a particularly popular term through its use to describe a series of songs released through the Disney Renaissance that had the main character singing about how they are unsatisfied with their current life, and what they are searching for. The term "'I Want' song" is believed to have been coined by Lehman Engel.
It's not really that there is a "formula" for these things, but I have learned over the years that pretty much any successful musical you can name has an "I Want" song for its main character within the first fifteen or so minutes of the show. I can think of exceptions, but frankly I feel that the lack of such a moment is a weakness in most of those cases. "Just Around the Riverbend" may not be a classic "I want" song, because the character doesn't really want anything that strongly until she meets John Smith, but it sets up her sense that she has another destiny to pursue than the one laid out for her by her father and society and her desire to go after it. The third number, "Mine Mine Mine", was basically supposed to introduce the antagonist, Ratcliffe, and what HE wanted, so that the central dramatic conflict could be established.
Placement within a musical
Musical 101 explains: "The Main "I Want" Song comes early in the first act, with one or more of the main characters singing about the key motivating desire that will propel everyone (including the audience) through the remainder of the show. In many cases, these songs literally include the words "I want", "I wish" or "I've got to". Classic examples include My Fair Lady's "Wouldn't It Be Loverly", Carnival's "Mira", The Sound of Music's "I Have Confidence" and "King of Broadway" in The Producers.
Bob Fosse said there were only three types of show songs from a director's point of view there were only: "I Am" songs – a song that explains a character/situation, "I Want" songs – desire/motivations, and New songs – A song that doesn't fit the other categories.
Beyond a musical
Schwartz also notes "I Want" songs are usually those which have a life beyond the musical they were featured in:
I don't think it's surprising that "I Want" songs tend to be among the most recorded -- they are often somewhat more liftable than other songs in the show (that is, they make sense outside the framework of the show) and they give the singer something to act. In classic terms, the job of an "I Want" song is not to move the action forward, but to set up the desire of the leading character that will drive the action for the rest of the show.
The term has retroactively been used to describe older "I Want" songs. In a top ten list of Disney, The Daily Dot ranked Robin Hood's "Not In Nottingham" as the best "I Want" song. The site also noted that these could be sung by antagonists, ranking The Hunchback of Notre Dame's "Hellfire" at number 5. FanPop listed "Part of Your World" from The Little Mermaid as the best song of this type. The WFPL article Great 'I Want' Moments in Musicals listed "Belle", "Somewhere That's Green", "Wouldn't It Be Loverly", "Lonely Room", "Corner of the Sky", and "Part of Your World". Schwartz has also written "I Want" songs for live action musicals, including "Corner Of The Sky" for Pippin and "The Wizard and I" for Wicked. In the musical Hamilton, the "I Want" song is "My Shot".