Three and a half years late. 
This was originally a story by Isaac Asimov. I don’t remember the content exactly, but I’ll reconstruct it as best I can.
Ray Silverstein was the human attache to Qikblamefar, an alien ambassador from the planet Sortibakenstrete. Since the name is rather long and unwieldy for a human tongue, aliens from Sortibakenstrete were called Foys. After a long and distinguished career as a diplomat, Qikblamefar had to retire due to declining health. He wished to return to the Foy world to die, but no space ship was fast enough to make it in time.
Foys had certain physical differences from humans, the chief being that Foys had five hearts. A doctor named Maude Ramirez specialized in alien diseases, and determined that Qikblamefar’s illness resulted from his hearts enlarging. She had wanted to obtain Qikblamefar’s hearts for study after he passed, but the Foy ambassador refused.
Ray knew the reason for Qikblamefar’s refusal stemmed from a Foy taboo, and looked for ways to change the ambassador’s mind. He learned that Foys had a strong belief in the power of music, stemming from a time when Foy warriors fought far from home. Their comrades would form a choir around the mortally wounded, and the power of their song would return the soldier back to his home. If Ray could arrange for a grand musical send-off, maybe Qikblamefar would consent to Dr. Ramirez harvesting his hearts.
Ray contacted Harold Fortenheimer, leader of the world famous Fortenheimer Choir. The singing group had performed in front of audiences on all seven continents and were beloved universally. Upon hearing Ray’s story, Harold Fortenheimer said he would be glad to arrange for the choir to perform at Qikblamefar’s send-off.
So, the Fortenheimer Choir, Dr. Ramirez, and other notables were put on notice to attend Qikblamefar’s death, which was due to happen any moment now. However, Ray still had not formally received permission from the Foy ambassador, and waited patiently for Qikblamefar to come to a decision.
The call finally came in. Ray saw Qikblamefar on the monitor screen in his deathbed, weak and barely able to speak. However, he had made his decision. “Give my big hearts to Maude, Ray. Dismember me to Harold’s Choir. And tell all the Foys on Sortibakenstrete that I will soon be there.”
A man walks into a nightclub with a beautiful girl on his arm
The show begins and the comedian comes out for his first show of the evening
The comedian says “A man walks into a nightclub with a beautiful girl on his arm
The show begins and the comedian comes out for his second show of the evening.
The show begins and the comedian says “A man walks into a nightclub with a beautiful girl on his arm”
Just then a man in the front row stands up and says “I think I’ve heard this before”
The comedian says “Well maybe you caught my first show of the evening”
The man says “No, I just walked in here”
The comedian says “Well it was a guy looked just like you walked in with a beautiful girl on his arm could have been your twin brother”
The man says “My twin brother’s dead”
The comedian says “What is this, a wake?”
The man says “I don’t have to stand for this”
And he stands up and he walks outside
And the comedian says “Are you out there? I can hear you breathing.”
The man says “I’m holding my breath”
The comedian says “Well I’m holding you wife”
Just then the man says “That’s not my wife”
And he walks back into the nightclub with another beautiful girl on his arm
“Who’s that lady I’m seeing you with” the comedian says
The man says “This is my wife. That other lady is my dead twin brother’s wife
You can take her if you want her”
And the comedian says “Not unless you say please”
Just then, a man walks into a nightclub with a tatoo of a beautiful girl on his arm eating elbow macaroni.
The comedian says “Is that girl from Italy?”
The man says “No just hungry”
Just then a man walks into the nightclub, he comes riding into the nightclub, on a pony, with a feather stuck into his hat
“What do you call that?” the comedian asks
“An entrance” the man says “But forget that”
Just give me a beer and give my pony a jockey”
The bartender says “I think that pony’s had enough already”
“Well make it a short jockey” the man says
“And while you’re at it give that lady’s lawyer some briefs”
The lady stands up and says “I can defend myself, your Honor”
And the lawyer says “But I’ll defend her honor, your Honor”
The judge says “Well on her or off her, make up your mind”
The comic says “Definitely on her, that’s the best offer I’ve had all day”
“Well take it or leave it” says the Judge
“Couldn’t we just drop it?” says the comedian
He says “You better drop leaflets before you bomb”
And the comedian says “I’m already bombing”
He says “Maybe it’s your material”
He says “You don’t think it fits?”
He says “Well it could be let out a little”
The comedian says “How much do you think it will cost me?”
He says “It’ll cost you an arm and a leg”
The comic says “Well listen, could you put it on the cuff?”
The tailor says “I’ll tell you what I’ll do. We’ll forget the leg and I’ll just charge you an arm”
And a beautiful arm it is
“OK” says the comedian and the tailor cuts off the comedian’s arm and gives him the suit
The tailor calls his girlfriend and asks her to go out on the town with him in order to celebrate
He calls on his girlfriend and gives her the beautiful arm as a gift
She wears it around her neck just like a stole and they go out on the town
The man walks into a nightclub with a beautiful arm on his girl
The show begins and the one-armed comedian comes out for his last show of the evening.
He does his act, and the audience stands up and gives him a hand
Ow!
Everybody who has a dog calls him “Rover” or “Boy.” I call mine “Sex.” He’s a great pal, but he has caused me a great deal of embarrassment.
When I went to city hall to renew his dog license, I told the clerk I would like a license for Sex. He said, “I’d like one too!”
Then, I said, “But this is a dog.” He said he didn’t care what she looked like. Then, I said, “You don’t understand, I’ve had Sex since I was 9 years old.” He winked and said, “You must have been quite a kid.”
When I got married and went on my honeymoon, I took the dog with me. I told the motel clerk that I wanted a room for my wife and me and a special room for Sex.
He said, “You don’t need a special room. As long as you pay your bill, we don’t care what you do.” I said, “Look, you don’t seem to understand. Sex keeps me awake at night.” The clerk said, “Funny – I have the same problem.”
One day, I entered Sex in a contest, but before the competition began, the dog ran away. Another contestant asked me why I was just standing there, looking disappointed. I told him I had planned to have Sex in the contest. He told me I should have sold my own tickets. “But you don’t understand,” I said, “I had hoped to have Sex on TV.” He said, “Now that cable is all over the place, it’s no big deal anymore.”
When my wife and I separated, we went to court to fight for custody of the dog. I said, “Your honor, I had Sex before I was married.” The judge said, “The courtroom isn’t a confessional. Stick to the case, please.”
Then, I told him that after I was married, Sex left me. He said, “Me too.”
Last night, Sex ran off again. I spent hours looking all over town for him. A cop came over to me and asked, “What are you doing in this alley at 4 o’clock in the morning?” I told him that I was looking for Sex. My case comes up Friday.
The entire Little Bunny Foo-Foo song is a shaggy dog story. Or a shaggy rabbit song.
So there once was this wasp that lived in a jungle. This was not your ordinary wasp though-he was smart, philosophical even. One day he finally got fed up with his repetitive, insignificant life and decided that he would leave his hive, his family, his entire close-knit wasp community and he would go out into the world and make something of himself, just like the humans do. So the wasp enrolls in school, and passes with flying colours. Remember, this is a very smart wasp. He gets his high school diploma in a little under 3 years, with a 4.0 GPA and all that snazz. After high school, believe it or not, the wasp gets accepted to Harvard. Harvard! This too proves to be no challenge for our hero, as he graduates in just two years, again a 4.0, on the Dean’s list, and all that snazz. Not to mention all the clubs and sports he was in-the newspaper, rowing, student government-and the fact that he was by far the most popular student on campus. Even his professors looked up to him.
He goes on to get two PhDs, and when he finishes his education, the wasp faces a bit of a dilemma. How does he apply his knowledge now? Where does he go from here? He decides to try out politics. After all, he was popular throughout school, did well in Harvard government. So he runs for mayor, and wins in a landslide. He greatly reforms the city, fixing virtually all its major problems. He runs for governor and again wins in a landslide. Two years later, the presidential election was coming up, and the wasp decides he might as well go for it.
Of course, he wins in the largest landslide in US presidential history. His presidency goes exceedingly well-he is loved by all parties, and has the highest approval ratings in history. He also finds the cures for cancer, AIDS, and broken hearts while in the White House. After 8 years (yes, of course he was reelected) the time has come for him to leave his office. Even his successor his saddened by the wasp’s departure, but they all know it’s what must be done. Back at his vacation home in California his first day after leaving office, the wasp looks back on his long and fruitful life. He realizes that he hasn’t been back to his hive at all since that first day he left. He suddenly feels a twang of guilt as he realizes how much he misses his parents and his little brother. So he heads back to the hive, looking more worn out than he remembers. He goes inside and greets his family, who are overjoyed at the sight of him. He talks about how his life has gone as his family listens in wonderment. Eventually he decides he is thirsty, so he decides to visit the old watering hole he remembered. Once he gets there though, there’s an extremely long line. He decides it’s worth the wait, so gets in line. One hour. Two hours. This is the slowest moving line he’s ever seen! Eventually he calculates that it could be a few days before he gets to the front of the line, so decides it’s not worth it. He decides to go get some cider to drink instead, but waddya know, another huge line of people waiting for cider! He remembers one other drinking area that never had a long line-fruit punch! So he decides to go get punch. He arrives, and lo and behold, there’s no punch line.
I hope nobody will take offense if I name this the worst collection of jokes, shaggy or otherwise, I’ve ever heard in my life!
One of my favorite shaggy dog jokes is the Lawrence Block short story “Cleveland in my Dreams” made into a 30 minute film! I’ve never timed myself, but I think it just takes a few minutes to tell it well enough to get a laugh. (PM me and I’ll tell it here.)
Here’s another I’ve mentioned at SDMB before:
A man walks into a pub in Ireland and orders three pints of beer. He takes them to a table and drinks alone, sipping in turn from each glass. Before long, he goes back to the innkeeper and asks for three more pints.
“Our Colleen would be happy to bring you your beer one pint at a time, sir. That way it will be cold.”
“Well, I’m drinking with me and me brothers. One’s in America; one’s in Australia; we’ll probably never meet again in this lifetime. But we always drink beer together!”
The guy became a regular in the pub. Everyone thought it was a sentimental story and greeted him with well wishes whenever he showed up. Until one day, when he ordered just two pints. 
The news got around, and the pub was a little quiet when the guy later walked over to order two more pints. The innkeeper shook his head, and said “we’re all grieving with you. We’re very sorry for your loss.”
The guy looked puzzled for a while, then smiled. “Oh no, I know what you’re thinking but me brothers are fine. It’s me! Doctor told me to stop drinking!”
(Typing it out and comparing its length with other jokes in the thread, I guess it doesn’t qualify as a “shaggy dog.”)
A one sentence one, From the Bulwer Lytton contest:
“Upon discovering that Miles Black, the famous phrenologist from Yorkshire, was going to take up yodeling to lonely goats in Bali, James White decided to balance four planks of wood on a beer keg and call it an abstract work of art in the style of a famous fourteenth-century architect, just going to prove that people will read any old garbage if they think there will be a good pun at the end of it.”
Most of the above stories have bland nothingburger endings, which I guess was the original thing about “shaggy dog stories”. There’s another genre of these that end with gruesome puns, often playing on some well-known phrase or expression, like “Tarzan stripes forever” or “People who live in grass houses shouldn’t stow thrones” or “If the foo shits, wear it”.
There’s a whole web site devoted to these.
Allen Funt used to feature these on Candid Camera from time to time.
ETA: One of the Asterix comics, I think it was “Asterix in Switzerland”, starts with a shaggy dog story, fully illustrated of course, that takes the first four pages until it leads finally to a bad pun.
A family takes their kid to see a magic show. The kid is so impressed he asks to see the magician after the show. While in the magician’s dressing room he notices the magician’s rabbit. The kid asks the magician if he can have it. The magician replies:
“Silly kid, that rabbit is for tricks.”
Okay, I found it on-line.
Oops, it wasn’t the first four pages, it’s just the first two pages. Story begins with elaborate two-page shaggy dog story.