Made famous by Alice in Wonderland, the practice of celebrating an unbirthday was something that Lewis Carroll/Charles Dodgson actually did in real life. Rather than give presents only on expected gift-giving days, Dodgson would surprise friends with gifts on other, random days.

In the Disney movie, the unbirthday discussion occurs during the Mad Tea Party. In the book, though, it's a conversation with Humpty Dumpty. Once again, philosophical and linguistic themes show through:

(Humpty informs Alice that the cravat he is wearing was an unbirthday gift from the White King and the White Queen. Alice says she prefers birthday presents, and Humpty rebuts this with the argument that you can get unbirthday presents 364 days a year, birthday presents, only one. He concludes triumphantly:)

"There's glory for you!" "I don't know what you mean by 'glory,'" Alice objected. Humpty Dumpty smiled contemptuously. "Of course you don't--till I tell you. I meant 'there's a nice knock-down argument for you!'" "But 'glory' doesn't mean 'a nice knock-down argument," Alice objected. "When I use a word," Humpty Dumpty said, in a rather scornful tone, "it means just what I choose it to mean--nothing more nor less." "The question is," said Alice, "whether you can make words mean so many different things." "The question is," said Humpty Dumpty, "which is to be master--that's all." Alice was too much puzzled to say anything . . .

The Unbirthday Song

Statistics prove, prove that you've one birthday,
one birthday ev'ry year.
But there are three hundred and sixty four
unbirthdays.
That is why we're gathered here to cheer.
A very merry unbirthday to you, to you.
A very merry unbirthday to you,
It's great to drink to someone and I guess that you will do.
A very merry unbirthday to you ¹

A very merry unbirthday
To me
To who?
To me
Oh, you

A very merry unbirthday
To you
Who, me?
To you
Oh, me
Let's all congratulate us with another cup of tea
A very merry unbirthday to me

chorus

Now blow the candle out, my dear
And make your wish come true
A very merry unbirthday to you²

____________________

As far as I'm concerned, this is one of the best songs in Walt Disney's Alice in Wonderland. I had the long-playing record and the accompanying book as a child, and I just about wore it out, listening and singing along. The original conversation about unbirthdays is between Alice and Humpty Dumpty, in Chapter 6 of Through the Looking Glass:

"I mean, what is an un-birthday present?"

"A present given when it isn't your birthday, of course."

Alice considered a little. "I like birthday presents best," she said at last.

"You don't know what you're talking about!" cried Humpty Dumpty. "How many days are there in a year?"

"Three hundred and sixty-five," said Alice.

"And how many birthdays have you?"

"One."

"And if you take one from three hundred and sixty-five, what remains?"

"Three hundred and sixty-four, of course."

Humpty Dumpty looked doubtful. "I'd rather see that done on paper," he said.

Alice couldn't help smiling as she took out her memorandum-book, and worked the sum for him:

                               365
                                 1
                               ___

                               364
                               ___

Humpty Dumpty took the book, and looked at it carefully. "That seems to be done right -- " he began.

"You're holding it upside down!" Alice interrupted.

"To be sure I was!" Humpty Dumpty said gaily, as he turned it round for him. "I thought it looked a little queer. As I was saying, that seems to be done right -- though I haven't time to look it over thoroughly just now -- and that shows that there are three hundred and sixty-four days when you might get un-birthday presents -- "

"Certainly,"' said Alice.

"And only one for birthday presents, you know. There's glory for you!" ³

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¹ (No author listed) http://www.stlyrics.com/lyrics/aliceinwonderland/theunbirthdaysong.htm
²Music: Sammy Fain / Lyrics: Bob Hilliard http://disneytickets0.tripod.com/alicelyrics4.html
³ Lewis Carroll, Through The Looking-Glass (And What Alice Found There), 1872.