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A few are referenced in the dub of Axis Powers Hetalia. Special mention goes to when England meets with his imaginary friends ("But let's not have any murders this go-around, shall we?") and the flashback episode where France and England, who are settling in the New World, find a crying Finland ("Did you get your head caught in the butter churn again?").
Yukari occasionally brings up a Love Letter incident involving Minamo from when they were in High School, which usually prompts Minamo to scramble to get her to shut up. Once, she sang "love letter" (In English) whilst flitting down the hallway, just before Minamo could relate a juicy detail about Yukari to the students. Watch the anime version of the scene, as it must be heard to be appreciated.
Another Noodle Incident (at least from Minamo's point of view) happens during a Beach Episode when she gets drunk and answers all the girls' questions about sex. In the morning, Minamo wakes up with a hangover and no memory of the previous night; then 11-year-old Chiyo asks her to clarify a few points... While she's digesting that, the other girls show up to formally thank her for the very educational lesson.
Edaniel has his own letter column on the Bizenghast website. In one letter he mentions that he once dated a human agent, who he later broke up with at the Mausoleums request. Why the Mausoleum wanted them to break up is never mentioned.
Rukia's first visit to the human world, which she can't even remember, is glossed over by Kiyone as soon as it's mentioned. What happened and why Rukia cannot remember it ever happening, isn't explained.
When Byakuya activates Senkei against Ichigo, Byakuya mentions in passing this is only the second time he's ever let someone see Senkei. Who was the first target of Senkei and why has yet to be explained.
Hisagi once brought a motorcycle back to Soul Society from the World of the Living. Something happened and it was confiscated. What happened has not been explained.
This exchange when Kallen and C.C. are holed up in the Chinese embassy:
Kallen: This reminds me of Aomori. C.C.: It isn't as bad as that was. At least everyone here is wearing their clothes.
An image released with a special edition DVD for the end of the series depicts C.C., Kallen, and another soldier running from searchlights while wearing nothing but towels, explaining this incident and rendering it no longer noodle.
It's also never explained why Rakshata calls Lloyd the "Earl of Pudding" though most fans make the obvious assumption
From Britannia's perspective, the Battle of Itsukushima which gave Tohdoh his title Tohdoh of Miracles. It marks Japan's only victory against Britannia during the invasion and is never explained how it comes about, just that the Britannian military still remembers it with bitter regret. Whether or not future fluff will reveal anything more on the subject remains to be seen.
Played with in the Cromartie High School anime when the rather upstanding Kamiyama refused to explain exactly why in the hell he's in a school full of Japanese Delinquents. However, he tells the audience several times that if they want to know they should read the manga (in which it is actually explained).
Almost nothing concerning Allen's time with Cross is outright shown or explained; Allen gets too traumatized remembering.
Allen once recalls an incident involving his mentor Cross Marian's stash of alcohol-filled chocolates. What happened after he ate them is never explained explicitly, but the incident pissed Cross off enough that he punished Allen big-time and forced him to promise to never touch a drop of booze again. In the "Kuro no Bansankai" Special Drama CD, however, we find out how Allen behaves when drunk: he gets very vicious and destructive, and when it wears off, can't even remember he was the one who caused all the massive property damage. As a result, Lavi is able to take a very, very good guess as to why Cross banned him from ever touching alcohol again.
When he was playing cards against Tyki and his friends, he mentions he had to gamble (and therefore cheat) to survive, and off to the side, we see something saying "a typical day", where we see only dialogue talking about Cross owing money.
There's occasionally reference to an incident where Izaya framed Shizuo for a crime. What crime that might be has never been elaborated upon.
To say nothing about their high school days. Shinra was apparently doing a lot of damage control:
Shinra: Other things happened too when I was in school. There was this "The Pool of Hell Opens - Chlorine Gas Everywhere!" incident, "Real Flesh Art in the Art Room" incident, "Cultural Festival Campfire Explosion" incident and of course "Field Trip to the Shambles - the Okinawa Arc"...during the three years when we were in Raijin, gasoline drums were rolled down from school buildings as part of the routine. Really, it's hard to believe how nice and orderly this school has become these days...
It has been mentioned that Aya has had many unusual exploits as student body president. One specific one apparently involved the Dean of the school's statue.
Hatori would often threaten to reveal Shigure's many incidents if he kept making an ass of himself (or those around him).
What did Pinako Rockbell do to earn the title "Pantheress of Resembool" in Fullmetal Alchemist?
In Get Backers, the reason for Takuma Fudou'sstalker obsession with Ban is vaguely explained by Ban to have started from Ban ripping Fudou's arm off during an incident in the past. It's never elaborated on (other than that it happened while Fudou was working as a raider). However, knowing Ban's arrogant, condescending attitude, along with Fudou's Ax-Crazy and depraved ways... it's not hard to guess the reason why it happened.
In Girls und Panzer, the volleyball team was disbanded under unknown circumstances during the previous academic school year, and its members, now second-years, join the tankery team in the hopes of reviving it. It's never revealed what happened to it, but it's possible that it's related to budget cuts and Oarai being on the brink of being shut down.
It's never explained what happened to Mayu that made him fatally ill and distrustful of adults.
The incident that turned class 2-4 (then class 2-3) into disaffected teacher-hating punks was not addressed in detail until near the end of the manga's run.
In chapter 46 of Katekyo Hitman Reborn!, Tsuna visits Longchamp's house. While there, Longchamp is attacked by a girl named Pantera (although he assures Tsuna that she's aiming for him, but has really bad aim). When Tsuna realizes this, he asks if something happened between the two of them, and Longchamp's Number 2 guy assures him that nothing happened and not to ask him about it again. Tsuna takes this as proof that something happened. Read the page here.
At one point this is averted just to then be played straight. After Motoko runs away from the Hinata House, we learn from her big sister that her hatred for men comes from a time when she caught them kissing. But she puts it as "since then, Motoko started to loathe men and turtles". One wonders what the poor turtles have to do with it, like Keitaro wondered.
The anime and manga briefly imply, though not outright stating, that her fear of turtles stemmed from fighting a demon that resembled a turtle.
In Lucky Star there are several scenes that begin with one character finishing a conversation talking about how something stinks. It's a bit of a running gag, but after seeing it mentioned two or three times it makes you wonder what smells so bad.
Much later, repeated mention is made of some apparently epic misadventures Negi had during Christmas and Valentine's Day. What happened then we can only guess at.
In the Mai-Otome anime, Shizuru and Youko recall a mishap that occurred during Natsuki and Mai's attempt at the survival exam. Shizuru notes that when she heard Natsuki's distress signal, she rushed to the scene as quickly as she could, only to find the two girls in a "most improper" position, at which point Natsuki angrily cuts her off and tells her to "forget about it" before she gets to the juicy parts.
In Naruto, Suigetsu is about to mention a past event that happened between Sasuke and Karin (some translations imply it was something she did to him), but Karin punches him in the face before he can elaborate. It was later revealed in a flashback — he saved her from a bear during the chunin exams, though why that's embarrassing is unclear.
The anime's Lovable Sex Maniac Brock, during the Orange Islands sub-arc, seems to legitimately settle down with the Island's female Pokémon Professor, actually leaving the group for a while to pursue the relationship. When he returns, it's clear that something happened to turn the relationship sour, but he refuses to elaborate, instead going into a deep, brooding, Blue with Shock depression whenever anyone even mentions the relationship or the Professor's name.
Subverted when it came to the origin of Dawn's Embarrassing Nickname, Dee Dee (Pikari in Japan). She handled her kindergarten's Plusle and Minun too roughly and the resulting shock made her hair sparkle.
A variation occurred where the Noodle Incident in fact happened immediately when it was first mentioned, but never elaborated on: In "The Battle of the Badge," Giovanni, while chewing out Team Rocket for supplying him with a Togepi (ironically, despite it being a rare pokemon, he had no interest in it whatsoever), he received a call from someone, only to react with shock and horror, and demanding when it happened, before departing quickly and having to abdicate his position as Gym Leader to the trio, before departing from a fake wall, and then telling his mystery pokemon (Mewtwo) that he has an emergency assignment for it before unleashing it. Exactly what happened was not really explained.
In that same episode, there was another variation. Gary had ten Badges currently before challenging Giovanni (more than the required eight; he did it to show off) which means there are at least three Gyms in Kano that Ash never visited. (Likely four, as Brock never mentioned Gary in his first appearance.) Information on these Gyms are never elaborated on.
Cilan and his incident with a Purrloin that was mentioned in one of the Best Wishes episodes. It made him terrified of all Purrloin.
In a similar manner, what made Georgia a Dragon Buster in the first place. All we know was that she lost a battle to a Dragon Master from the Village of Dragons. We don't know who defeated her, how she was defeated, or anything else. However, given that it made her go on a revenge plot to defeat every Dragon-type she saw, and how she reacted when Iris' Axew learned Outrage, it's safe to say that it caused a lot of trauma for her.
Revolutionary Girl Utena has Tsuwabuki's secret Nanami diary. The end of the episode of the same name leaves the audience wondering... "What is Plan 24?" Not to mention Plans 3 through 23, really.
Lina Inverse references the horrible punishments her sister Luna used to put her through quite a few times, even though they are never actually explained. It leads one to wonder what she really could do, granted that Lina could put up with being electrically shocked while saying it was nothing compared to that torture. That, and Lina constructed a pyramid atop herself in order to merely get away from a letter sent by Luna and just the mere reference to her causes Lina to fly into a mortal panic.
One of the less harsh punishments Lina suffered involved a whole lot of slugs, a fact which she tearfully reveals after a near-total fear-induced breakdown in the second season.
An unspoken Noodle Incident is whatever it was that caused Naga the Serpent, Lina's self-proclaimed rival (and whom Lina publicly called "Goldfish Poop" in the novels, due to her habit of following Lina), to stop following Lina by the time the anime started.
Uchouten Kazoku's dialogues are sprinkled with little Call Backs to funny or outrageous histories the characters know all too well but the audience really doesn't.
In one episode of Yes! Pretty Cure 5, Nozomi decides to help out Urara's first show by taking up the show's lead. Rin's quick to point out that the last time she tried acting, she ruined the Drama Club and got kicked out in two days. Nothing's ever said as to how she did it, though.
In Attack on Titan, Jean mentions that Bertolt's "artistic" sleeping positions were so bad, the other trainees used them to predict the weather. What that means is never elaborated on, leaving creative fans to come up with their own ideas.