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Of unknown origin, the phrase shaggy-dog story designates a long rambling joke ending in a deliberate anticlimax, such as an absurd or irrelevant punchline.

The following, on the origin of the phrase shaggy-dog story, is from Allen’s Dictionary of English Phrases (London: Penguin Books Ltd, 2008), by the British lexicographer Robert Allen [s.v. shaggy]:

The expression probably arose from the fact that a shaggy dog was a common subject for stories of this kind. Apocryphal but entertaining anecdotes have been put forward to explain the origin more precisely and more colourfully. These mostly centre on lost dogs whose owners advertise for their recovery in the newspapers. Various responses follow: all dogs of the area that fit the description are rounded up and the shaggiest is chosen, a reader takes pity on the owner and proceeds to replace the dog with one that is even shaggier, and so on. Shaggy-dog stories indeed.

—However, cf., below, quotation 6, in which the author claims that the phrase shaggy-dog story did not originally refer to jokes about shaggy dogs.

These are, in chronological order, the earliest occurrences of shaggy-dog story that I have found:
—Note: Apparently, this phrase (which seems to originated in U.S. English) was already so well-established by 1914 that none of the persons using it in the first five quotations felt the need to explain it:

1-: From History and Personnel of Nu Chapter: A Resume, Chiefly Concerning Matters Personal, by J. Marius Scammell, published in The Tomahawk: Official Publication of Alpha Sigma Phi (Menasha, Wisconsin, USA) of February 1914 [Vol. 11, No. 2, page 89]:

At our banquets, we still have the “shaggy-dog story” retained, and with it the old tradition of how the Atherton Club once nearly entertained Carrie Nation. That was in Nu’s vigorous childhood.

2-: From a review of Lightnin’, a comedy play by the U.S. playwright Winchell Smith (1871-1933) and the U.S. playwright and actor Frank Bacon (1864-1922), produced at the Gaiety Theatre, Broadway, New York City—review by Emory B. Calvert, dated New York City, Saturday 21^st September 1918, published in the South Bend News-Times (South Bend, Indiana, USA) of Sunday 22^nd September 1918 [Section 2: page 6, column 1]:

The next scene reveals the superior court at Reno, which really isn’t so darn superior, as the fellow said about the shaggy dog. (Note—Shaggy dog story will be mailed on application to editor.)

3-: From an article about a partnership arrangement whereby John H. Payne and Charles J. Lilley became part owners and associate publishers of the Sacramento Union—article published in the Sacramento Union (Sacramento, California, USA) of Thursday 23^rd January 1930 [page 3, column 1]:

Payne is a native of Nebraska, married, and has three sons. He answers to the nickname of Shaggy, because of a shaggy dog story he has made famous.

4 & 5-: From one of the unconnected paragraphs making up the column By the Way, by ‘Whitey’ Parcells, published in The Waukegan News-Sun (Waukegan, Illinois, USA):

4-: Of Thursday 14^th March 1935 [page 12, column 3]:

Wonder how many of the letter men at Lake Forest College have heard the “shaggy dog” story?

5-: Of Wednesday 24^th April 1935 [page 14, column 3]:

Floyd Brown is taking over the Lake Forest baseball team because he wants to have a new group to listen to the “shaggy dog” story . . . He feels that it would be a shame for the younger generation of Lake Foresters to grow up without hearing that classic . . . “Speed” and “Joe” Lindenmeyer, “Jake” Hansen and Bill Davis are a few of those who have promised to be around and aid Floyd in giving out the details of that intricate tale.

6-: From Don’t Laugh Now: If you’re so highly evolved you’re slightly screwy, you may get a kick out of these, by the U.S. author Joseph Chamberlain Furnas (1905-2001), published in Esquire: The Magazine for Men (New York City, New York, USA) of May 1937:

[page 56, column 1]:
THE majority of any given group of people are probably all too sane. One of the more sporting ways of finding out which ones are not is to try shaggy-dog stories on them. Take the story of the cake as a sample:
One morning a morning-coated, top-hatted man came into a famous New York confectioner’s, ordered a special cake to be made in the shape of an S—and said he would call in to inspect it the next day. Expense was obviously no object, so the confectioner had a special S-shaped pan made up at the tinsmith’s and baked a beautiful cake, lavishly decorated. When the customer arrived next morning, however, he was not at all satisfied: “This is a block S,” he said; “I wanted a script S.” The confectioner apologized, ordered another pan and baked another cake. In the morning the customer was still unhappy about it: “No, this won’t do,” he said. “The curlicue is cut off. I want a large and graceful curlicue, do you understand?” The confectioner, tactful and patient through it all, ordered another pan with an elaborate curlicue and baked another cake. This time the customer was delighted—the cake was perfect. “I’m so glad, sir,” said the confectioner. “Where shall we send it, sir?” “Never mind,” said the customer, “I’ll eat it here.”
When confronted with that one, two of a job lot of listeners will laugh hollowly and briefly, unmistakably bewildered and yet trying to be polite. Two more, probably forthright young women, will say, frowning attractively: “But I don’t see anything to laugh at.” Certain kinds of middle-aged men will turn away in silent contempt. But those members of the party who are sufficiently matured spiritually to be slightly nuts will double up and whoop, regardless of the feelings of the uninitiate.
[page 236, column 3 & page 237, columns 2 & 3]:
They say they are known as shaggy-dog stories because the story of the shaggy dog was the first of the lot to become popular. That can’t be true, since the shaggy dog, besides being a poor specimen, seems to have appeared fairly recently and, to the writer’s certain knowledge, a couple of the classic examples go much farther back. But “shaggy-dog” will have to do because there is no other label which fits at all neatly. […]
Addicts sometimes express a fondness for the shaggy-dog story, however, so it should probably be included as clinical datum. An advertisement appears in a New York paper offering a £ 500 reward for the return of a certain large, white shaggy dog, marked thus and so, to an address in a London suburb. A New Yorker who has just picked up a big white shaggy stray with the indicated markings, immediately takes ship for England with the dog, goes to the advertised address and rings the doorbell. A man opens the door. “You advertised about a lost dog,” says the American, “a shaggy dog.” “Oh,” says the Englishman coldly, “not so damn shaggy” and slams the door in the American’s face.
[…]
The point in Shaggy-Dog stories may well be that, although heedless of sense, they are not nonsense. Instead they insist on making the logic of a situation impose itself ruthlessly on the hearer, who tricked himself into being surprised by making normal reflex shortcuts between an opening situation and the sort of conclusion one would expect in normal life.