This very famous question comes from Alice’s Adventures in Wonderland (by Lewis Carroll, Chapter VII, “A Mad Tea Party”):
The Hatter opened his eyes very wide on hearing this; but all he said was “Why is a raven like a writing-desk?”
“Have you guessed the riddle yet?” the Hatter said, turning to Alice again.
“No, I give up,” Alice replied. “What’s the answer?”
“I haven’t the slightest idea,” said the Hatter.
“Nor I,” said the March Hare.
Alice sighed wearily. “I think you might do something better with the time, “she said, “than wasting it in asking riddles that have no answers.”
(Here is a full text of this chapter, in case you wish to see the context.)
From Martin Gardner’s Annotated Alice, we learn that Carroll did not intend that there be an answer. In fact, Carroll wrote in the preface to the 1896 edition (I’ve emphasized the last sentence):
Enquiries have been so often addressed to me, as to whether any answer to the Hatter’s Riddle can be imagined, that I may as well put on record here what seems to me to be a fairly appropriate answer; vis: “Because it can produce a few notes, tho they are very flat; and it is never put with the wrong end in front!” This, however, is merely an afterthought; the Riddle, as originally invented, had no answer at all.
Here are some more answers, all documented in The Annotated Alice.
Because the notes for which they are noted are not noted for being musical notes.
Because Poe wrote on both.
Bills and tales (tails) are among their characteristics.
Because they stand on their legs, conceal their steels (steals), and ought to be made to shut up.
by Sam Loyd, American puzzle maven. The Poe answer is my favorite.
Because there’s a b in both.
Because there’s an n in neither.
by Aldous Huxley. In a philosophical manner, he responded to the nonsense question with nonsense.
Because each begins with e.
by James Michie.
Both have quills dipped in ink.
by David B. Jodrey, Jr.
Because it slopes with a flap.
by Cyril Pearson.
Because it can produce a few notes, tho they are very flat, and it is nevar put with the wrong end in front.
by Lewis Carroll. This is an updated answer, intentionally misspelling the word “never” as “nevar”, which is “raven” spelled backwards. The editor, however, mistook this for a typo and “fixed” it to “never.” Carroll died soon after, and so, “[w]hether Carroll was aware of the damage done to his clever answer is not known” (Gardner).
Because without them both Brave New World could not have been written.
by Roy Davenport.
Because one has flapping fits and the other fitting flaps.
by Peter Veale.
Because one is good for writing books and the other better for biting rooks.
by George Simmers.
Because a writing-desk is a rest for pens and a raven is a pest for wrens.
by Tony Weston.
Because “raven” contains five letters, which you might equally well expect to find in a writing-desk.
by Roger Baresel.
Because they are both used to carri-on de-composition.
by Noel Petty.
Because they both tend to present unkind bills.
by M.R. Macintyre.
Because they both have a flap in oak.
by J. Tebbutt.
Because it bodes ill for owed bills.
Because they each contain a river—Neva and Esk.
by Francis Huxley.
That’s all from The Annotated Alice: The Definitive Edition, and they are all very good. Here are some more I’ve found online and elsewhere:
Because a raven makes no sense, and so does a writing desk.
Because neither requires the other.
by Linus Connell.
Because neither one is made of cheese.
(Cannot determine origin of saying.)
Because you can baffle the billions with both.
Because they both stand on legs.
Because you cannot ride either one of them like a bicycle.
Because neither one of them is made from aluminum.
Because the raven wanted to be.