Crazy Memory

How Homer Simpson remembers the last town hall meeting.

I used to be an Asian Boy Scout! Wasn't that a weird week?

—Chester A. Bum, Up review

Bart: Is that story true, Grampa?

Abe: Well, most of it. I did wear a dress for a period in the forties. Oh, they had designers then!

—The Simpsons

This trope is when a character reminisces about something so outlandish and wild, it probably isn't true. Perhaps this character is out and out lying in an attempt to be interesting or shocking, or they might just be crazy enough to believe their crap. On occasion, if nobody else believes this person, it may actually prove to be true. In cases such as these they are quickly forgotten and never brought up again. Typically, this trope is played up for comedic effect and will often occur in lonely old people. Adventurers are also prone to these as they like to brag about their past adventures. Also common amongst the Cloudcuckoolander.

Compare Noodle Incident, Cutaway Gag, Non Sequitur Scene, and Imagine Spot.

Examples of Crazy Memory include:

Anime and Manga

Used in the Pokémon anime: Cabernet recounts her first encounter with Cilan: he thrashed her in a Gym, then he and his Pansage loom over her with their rape faces on, much to the horror of poor little Cabernet and her Oshawott. Arguably subverted, since Cilan acted rather unsettling during their rematch. Doesn't help that he's also a Memetic Molester.

Film

In Austin Powers: International Man of Mystery, Dr. Evil describes his father as this.

Dr. Evil: He would make outrageous claims, like he invented the question mark.

Happens during the riffing of Pirates of the Caribbean

Bill Corbett: I don't blame her, I'd rather jump off a cliff than marry a man in a tri-corner hat.
Kevin Murphy: That's experience talking isn't it. Really brings that lost year in colonial Williamsburg into focus.
Bill: We shan't speak of that.

- Speaking of which, Jack Sparrow actually has a couple of moments like these throughout the series. Most notable is an ending to a story he's telling two guards where he claims "and then they made me their chief."
  - Was that before or after he built a raft out of a couple of live sea turtles and a rope made from his own hair?
The entire point of Secondhand Lions. Great-uncle Hub's story is outlandish and wild about being in the French Foreign legion, fighting a sheik for the love of a princess, and all that sort of nonsense. The ending scene shows that it was all true.

Live Action Television

The entire history of the Janitor from Scrubs. If you think you know something about him, you don't, or it is disproved later and re-proven after that.

Janitor: My father died [when I was young].
JD: I met your father!
Janitor: You met a man.

- Also:

JD: Was any of that true?
Janitor: I don't know. Someone would have to read it back to me.

- When the janitor tries to grow facial hair:

JD: It looks like tiny hamsters died all over your face!
Janitor: That happened once.

The Twilight Zone episode "Hocus-Pocus and Frisby" was about a man who told outrageous lies to his friends about his past. It comes back to bite him when he's abducted by aliens who have no concept of lying. Fortunately, the one thing he's really good at - playing the harmonica - turns out to give them excruciating pain.
The first season of Two Guys, a Girl And A Pizza Place had a recurring character of a crazy patron who would tell stories about his life that were all actually plots of popular movies. Then he was Put on a Bus.
Captain Jack Harkness in Torchwood does this often in the form of references to relationships with historical characters (which could very well have actually happened since he's a time traveler and he's taken The Slow Path) and impossible situations like being pregnant (which we really, REALLY hope didn't actually happen.)
In Doctor Who, the Doctor will often slip in references to stories which may or may not involve Michelangelo, arrows, alien super-computers and pepper-pots with plungers. Random bystanders will think he's a madman. Well, he is a madman, but his stories are probably all true. Also, he's second only to Captain Jack at this point in terms of relationships with historical figures.
- Depends on how you define "relationships", really.
In one episode of It's Always Sunny in Philadelphia, Frank tries to pass off more than one story from John Rambo's life as something that happened to him. No indication is given that he doesn't genuinely believe these things happened to him.
Beans from Even Stevens was a lot like this.
Future!Ted from How I Met Your Mother occasionally had some of these, which were usually implied to be him simply misremembering things or being unable to recall what happened outside of his biased perspective, leading to the events of the episode suddenly veering into totally surreal territory for a few minutes. Usually prefaced by "Kids, I swear this is what happened next..."
- Barney often claims to have done some pretty crazy things. On the one hand, he does lie/exaggerate a lot, on the other hand...he is Barney so really anything's possimpible...

Newspaper Comic

Bloom County. Milo sometimes did this, but about Binkley's life, not his own. Note that Milo and Binkley are both children.

Milo: Tell me, Binkley, if you had your life to live over again, would you do everything the same? Would you have left home to join those Bohemian resistance fighters? Would you have married that Turkish leopard smuggler? Would you have spent those twenty years chasing white tigers in the mountains of Ceylon?

Stand-Up Comedy

The late Dave Allen frequently gave different reasons on how he lost one of his fingertips in order to derive material from his injury; the reasons included his brother accidentally bit it off, he maimed himself to avoid war, it was stuck in a bottle of whiskey that eventually dissolved it and he wore it down by obsessively brushing dust off himself (to name a few).

Web Original

As the above quote indicates, Chester A. Bum of Bum Reviews has a lot of these. He will often talk about times in his life where crazy and outlandish scenes from the movies he saw were actually like things that happened to him. Expect the line "except, instead of (insert activity here), I just did drugs" to follow many of these statements.
"I got stabbed in the part of my brain that produces flashbacks."
Red vs. Blue has Caboose who couldn't remember anything correctly to save his life, and Donut who's probably aware that he's making things up, but we never get any proof.
Solomon "Duke" Todd of Shadow Unit is prone to these, though it can be difficult to tell when they're real memoried and when he's just making them up...

Western Animation

Subverted and parodied in Futurama, twice. In the episodes "Fry and the Slurm Factory" and "A Clockwork Origin," Professor Farnsworth is declared crazy and everything he has just said has been lunacy. In retaliation, he begins ranting and shouts "and he's my uncle" pointing to the much younger character, Fry. This is actually true, as Fry comes from the distant past and is Farnsworth's great great great great great etc uncle (and his great great great great great great great etc grandfather. However, nobody believes him, writing him off as nuts.
The Simpsons: Many of Abe Simpson's ramblings fall under this.
- Homer's memory of Marge's comments at town hall in "$pringfield".

Homer: ...And so I said to the President, "Get this..."

Pretty much every Manatee Gag on Family Guy is one of these, where a character will say something like "This is just like when I..." before cutting away to a crazy situation that almost certainly couldn't have happened.
Arnold's grandfather on Hey Arnold! spouts these frequently. At one point, Arnold calls him out on a particularly outrageous war story:

Arnold: Grandpa, you did not fight Adolf Hitler. You're making this all up.
Grandpa: Heh heh heh. Okay, you got me. I made that part up. Pretty funny, huh? ...It was Goebbels.

Roger in American Dad has these on a regular basis. Of course, he's a 1600-year-old alien Master of Disguise who becomes the mask on a regular basis, so no one really knows how many of them are true. Roger included.
- Much like the Family Guy example above, American Dad will have characters constantly think back to outrageous situations as a lead in to a Manatee Gag.
Izzy on Total Drama may be lapsing into this constantly, but it would be hard to tell—she openly admits that some of her mad stories are just lies, while at least a few others seem to actually be true.
Gir from Invader Zim tends to fall pray to these. Especially when he is asked to remember something important.
During the My Little Pony Friendship Is Magic episode "The Cutie Mark Chronicles", the Cutie Mark Crusaders ask the Mane 6 about how they acquired their marks. Pinkie Pie explains how she grew up on a "rock farm", where there were "no laughing, no fun, only rocks" and she had to push rocks around all day, until the sonic rainboom made her feel joy and she threw a spontaneous party for her family. The scene cuts to the present and she finishes with "...and that's how Equestria was made!" much to the complete bafflement of the CMC. The fanbase is somewhat divided on the issue of whether her story is true or a complete fabrication.