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In Lewis Carroll's Alice series, the Mad Hatter says a riddle, Why is the raven like a writing desk? There isn't any answer to this riddle in this book. Why does the hatter say this and does this have any answer?

The author did not originally intend there to be an answer, but he was eventually badgered into producing one. According to this Gizmodo article:

The unanswerable riddle has been answered, though, and has been answered for many years. Lewis Carroll himself wrote the answer, after being badgered by people nonstop since the book's original publication. He said that, in the original book, there was no answer. To end the pain of ceaseless inquisitive fan letters, though, he went ahead and thought up an adequate response that he put in preface to later editions. Carroll's answer to why a raven is like a writing desk? "Because it can produce a few notes, tho they are very flat; and it is never put with the wrong end in front!" I'm sure your thighs are now sore from the repeated slapping they took after you read that line. Originally, it was supposed to be a little funnier than that. Carroll spelled 'never,' as 'nevar' — 'raven' spelled backwards — but a proofreader erased the inverted pun before it was published.

Lots of other answers have also been suggested. That article also points out two of the other famous third-party answers:

The unanswered riddle, which many people were exposed to in their formative years, got under people's skin. In their attempt to adequately extricate it, they've come up with answers. A satisfying, but meta, answer is, "Poe wrote on both," given by puzzle enthusiast Sam Lloyd. More in the spirit of the nonsense genre, Aldous Huxley ventured, "Because there is a 'b' in both and an 'n' in neither." Beautifully bizarre.

The riddle has no (canonical) solution.

The author, after having been repeatedly questioned on the subject for nearly two years after the original work was published, eventually came up with an answer for the updated edition which they included in the preface, reproduced below from the 150th Anniversary edition. Note that there is no answer, merely one that the author devised post-facto

PREFACE TO THE EIGHTY-SIXTH THOUSAND

Enquiries have been so often addressed to me, as to whether any answer
to the Hatter's Riddle (see p. 59) can be imagined, that I may as well
put on record here what seems to me to be a fairly appropriate answer,
viz., "Because it can produce a few notes, though they are VERY flat;
and it is nevar put with the wrong end in front!" This, however, is
merely an after-thought: the Riddle, as originally invented, had no
answer at all.

_{^{Alice's Adventures in Wonderland and Through the Looking-Glass: 150th Anniversary Edition}}

Various other wits have noted the lack of a true answer and offered their own opinions.

LEWIS CARROLL himself proposed an answer in the 1897 final revision of
Alice's Adventures. "Because it can produce a few notes, though they
are very flat; and it is never put with the wrong end in front!" The
early issues of the revision spell "never" as "nevar", ie "raven" with
the wrong end in front.

Martin Gardner, in More Annotated Alice (1990)
gave two possible answers, sent in by readers: "both have quills
dipped in ink" and "because it slopes with a flap". In 1991, The
Spectator held a competition for new answers, among the prize winners
were: "because one has flapping fits and the other fitting flaps";
"because one is good for writing books and the other better for biting
rooks"; and "because a writing desk is a rest for pens and a raven is
a pest for wrens".

(Dr) Selwyn Goodacre, Editor, Journal of the Lewis Carroll Society,
Swadlincote, Derbyshire.

_{^{Any possible solutions to the Mad Hatter's conundrum: Why is a raven like a writing-desk?}}

The same 'Guardian Newspaper - notes and queries' page contains an extended quote from John Fisher's The Magic of Lewis Carroll.

JOHN FISHER, in his book "The Magic of Lewis Carroll" (Thomas Nelson
1973, Penguin 1975), quotes Carroll's own answer, supplied in a
preface to the 1896 edition of "Alice in Wonderland": "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 be a fairly appropriate answer, viz: 'Because it can
produce few notes, tho [sic] 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." Fisher also quotes Sam Loyd's solution, in his posthumous
"Cyclopedia of Puzzles", published in 1914: "The notes for which they
are noted are not noted for being musical notes." Fisher continues:
"Loyd also reminded the world that 'Poe wrote on both' and that 'bills
and tales are among their characteristics.'"

_{^{Any possible solutions to the Mad Hatter's conundrum: Why is a raven like a writing-desk?}}