"We can't bust heads like we used to. But we have our ways. One trick is to tell stories that don't go anywhere. Like the time I caught the ferry to Shelbyville. I needed a new heel for m'shoe. So I decided to go to Morganville, which is what they called Shelbyville in those days. So I tied an onion to my belt, which was the style at the time. Now, to take the ferry cost a nickel, and in those days, nickels had pictures of bumblebees on 'em. 'Gimme five bees for a quarter,' you'd say. Now where was I... oh yeah. The important thing was that I had an onion tied to my belt, which was the style at the time. You couldn't get white onions, because of the war. The only thing you could get was those big yellow ones..."
—Abraham 'Grandpa' Simpson, The Simpsons, "Last Exit to Springfield"
Gender Inverted in Servant × Service; where it is the old women that likes to engage in long ramble to the welfare office staff. The only named character of this type, Mrs Tanaka, really loves to complain about her daughter-in-law to Miyoshi.
Early in Beetlejuice, Adam runs into his hardware store, saying hello to the old barber next door, who starts talking - when Adam leaves, the barber is still talking...
AbrahamLincoln has a habit of breaking into anecdotes that sometimes don't have any relevance to the topic at hand. Other times they're quite calculated to produce an effect.
Mr. Wojakowski in Connie Willis' Passage has a tendency of rambling off into WWII stories when he should be talking about his Near-Death Experiences. Then again, his WWII stories are mostly made up...
Mark Twain's The Celebrated Jumping Frog of Calaveras County. The entire story is this. The narrator is looking for an acquaintance named Leonidas W. Smiley asks an old man named Simon Wheeler about him. Wheeler says he doesn't know Leonidas but does know a Jim Smiley. He then proceeds to tell the narrator the story of the jumping frog. When that story ends, Wheeler tries to tell the narrator another story about Jim's one-eyed crow as the narrator attempts to escape.
The outdoor humorist Patrick McManus has discussed this trope in some of his short stories. And in later years, offered up straight-up examples of it.
Purdy in the Warrior Cats series. Fans love it, characters get annoyed by it.
Purdy: Do you ever miss hunting? Mousefur: As much as you would miss talking if your tongue fell out!
In 1984, when Winston Smith asks an elderly prole what life was like before the Party, he's unable to get a straight answer because the man keeps lapsing into these.
Everybody in Joan Hess's Maggody mystery series talks like this, young or old. This makes questioning people insanely frustrating for Chief Hanks, or anyone else who wants a straight answer for that matter.
Cliff does this sometimes in The Cosby Show, such as his lecture about the use of the line "Throw down that box" in his favorite Western.
In Game of Thrones Grand Maester Pycelle launches into a lengthy reminiscence about all the kings he's served. To make Petyr, who Roz is spying for, think he's going senile and declare his loyalty.
President Bartlet on The West Wing had a tendency to do this, though it was more about showing off his personal knowledge than encroaching senility.
A courtroom sketch on Monty Python's Flying Circus had a little old lady called in as a witness, whereupon she promptly starts rambling about nothing in particular. When the lawyer realizes she's not going to let him get in a word in edge-ways, he has her removed from the court, still rambling. ("Don't you talk to me about bladders, I said...")
"Old Blevins" by The Austin Lounge Lizards is the tale of young man trapped in a bar listening to Old Blevins deliver one of these. He realises that if he does not return home and patch things up with his girlfriend then he is in danger of turning into someone just like Old Blevins.
'Rambling Jack Elliot' earned his nickname due to this tendency. He wasn't old at the time, but nowadays...
EverQuest had an NPC called Old Man McKenzie. The official description of him is: "Old Man McKenzie, a frequent patron of the taverns in the Plane of Knowledge, thinks you adventurers have it too easy these days! Back in his day they didn't have all this fancy armor and magical weaponry, they relied on their wits and not a little luck to survive! Think you've got what it takes to survive in McKenzie's Gold era?"
Also Grumpy Old Man Victor Kudo from the third game, who, while not as bad, occasionally has trouble getting to the point in his testimony, and in one questioning sequence, the only way to get information out of him is to politely sit through his rambling.
The Legend of Zelda: Majora's Mask: Listening to such an old lady ramble on (and then fall asleep as she does so) is a way to skip ahead in time if you don't have the Song of Double Time, though if you manage to stay awake and listen to the whole story, you earn two Pieces of Heart out of the ordeal. The nice thing about it is that the stories she tells are actually kind of interesting and provide background on the story.
In Anachronox, this is one of the party member's explicit abilities.
Old Man Andrew of Mega Man Zero lapses into this. Every so often he asks you questions to make sure that you're paying attention. Answering correctly is required to get certain collectibles.
Kingdom of Loathing has Grandpa Sea Monkee. After finding him as part of the Sea quest line, you can ask him about a number of topics, most of which provoke a long, confused, and tangent-filled anecdote about Grandpa and his childhood friends.
Borderlands 2 has one in The Horrible Hunger of the Ravenous Wattle Gobbler DLC from Grandma Flexington, the grandmother of Mister Torgue. Listening to her extremely long-winded monologue is a sidequest and a test in patience since not only can you not move away, but she'll even ask you a question to see if you were listening. Failing any of them automatically causes her to fail the quest, forcing you to start the whole thing over. Afterwards, there's a Raid Level version of the mission, where she goes on an even longer monologue, though she'll send you on a fetch quest at some point.
Played for humor and heavily lampshaded in Danny Phantom at the end of "Doctor Disorders", where Tucker gets stuck staying in a hospital room with an old man explaining how many things they didn't have when he was growing up. All done in the most patronizing tone imaginable.
Grandpa Reg in Phineas and Ferb is prone to these. His not-so-old son, Phineas and Ferb's dad Laurence, seems to have inherited the tendency (although he's less likely to completely forget what he was talking about).
Played to a T on Animaniacs in the Warners short "Chairman of the Bored". Ben Stein plays a late middle-aged dullard named Francis Pumphandle ("But everyone calls me Pip"), who upon meeting the Warners at a party begins a long-winded, rambling anecdote involving Bob Barker, cheese balls, and orchids, and remains firmly oblivious to their increasing desperation to shut him off or escape. Naturally, once he does finish his story and leave, they beg him to come back.
In one episode Grandpa Wolfe shuffled along with his walker along the sidewalk in front of Rocko's house, rambling on and on... right up through nightfall.
In another episode Rocko, Heffer and Filbert go on a fishing tour with "Two-Patch" Crappie Jack (who's got wooden arms, legs, and even eyes). Crappie Jack is so busy with his rambling sailor anecdote that he fails to notice he's sailed off in the wrong boat, leaving Rocko and friends to fend for themselves.
He let out a scream 'twas heard in Davy Jones' locker! And Mickey Dolenz's locker, too, and Peter Tork's locker! All The Monkees had lockers.
In Rugrats, Lou Pickles (Tommy's grandfather) often does this. He also seems to have a strange fixation on the number fifteen while telling these tales.
"I've actually got an interesting story about that, although I guess it's not so much interesting as it is long... and boring..."
In another episode, Abraham let Homer have one of these.
"The year is 1946. In a war torn world, a single flower blooms, and that flower is an angry Japanese monster named Godzilla. Fortunately, there was one man who could help... Colonel Tom Parker. The Colonel took this monster, cleaned him up, and put him on stage under the name "The Rolling Stones." The first concert was a sellout with many many many people eaten, but those that survived RAVED about the undeniable harmonies and brutally honest lyrics of what they had just seen... And out of the wreckage of that concert crawled the woman who would later give birth to me... Not once, but FOUR times, because in those days, if they didn't like the way you looked, they would send you back in to bake a little longer..."