An unbirthday (originally written un-birthday) is an event that can be celebrated on any day that is not the person's birthday. It is a neologism coined in Lewis Carroll's Through the Looking Glass, giving rise to the song "A Very Merry Unbirthday to You" in the 1951 Disney animated feature film Alice in Wonderland.
In Through the Looking Glass, Humpty Dumpty is wearing a cravat (which Alice at first mistakes for a belt) which he says was given to him as an "un-birthday present" by the White King and Queen. He then has Alice calculate the number of unbirthdays in a year.
In the film Alice in Wonderland, Alice stumbles upon the Mad Hatter, the March Hare and the Dormouse having an unbirthday party and singing the song "A Very Merry Unbirthday to You" (music and lyrics by Sammy Fain and Bob Hilliard). Alice at first doesn't realize what an unbirthday is: when the Mad Hatter explains it to her, she realizes it is her unbirthday as well, and receives an unbirthday cake from the Mad Hatter. The scene from the film combines the idea of an unbirthday introduced in Through the Looking Glass with the "Mad Tea Party" described in Alice's Adventures in Wonderland.
The unbirthday party is also the subject of a 1951 comic released to coincide with the film. The comic version is substantially longer (32 pages) than the scene in the animated film, and has Alice being invited to the unbirthday party of Tweedledum and Tweedledee (who are not actually present at the unbirthday party). Humpty Dumpty is a character in the comic version, although not in the manner in which he appears in Through the Looking Glass.