AIR had just about got to a bad ending, halfway through the series — then it cuts to a few episodes of backstory in the feudal era, and after that, tells the whole story again from before Yukito came to after the show cut off, from the point of view of the crow. It's all the story of Yukito and Misuzu all along, though, in their different incarnations — yes, even the crow.
With all the gory and tragic H-games and animes from the Nasuverse, no one was expecting a silly and zany OVA like Carnival Phantasm.
Patlabor did this more times than you can count. Including: An episode revolving around ordering lunch for the entire crew, another about hunting for pearls in the albino alligator-infested sewers, TWO stories about alternate reality versions of the crew, and a lower decks episode about a rebellion by the maintenance staff.
The anime ends its chronological last episode with "Someday in the Rain," an episode the original novels' author wrote for the series. The episode is just a slow, meandering look at the activities of the SOS Brigade while narrator (and Deadpan Snarker) Kyon is out of the room. The main theme of the episode was "The SOS Brigade without Kyon", and therefore quite a lot of camera time is spent in one position, watching Yuki read. The reason for how this episode was done was to distinguish from usual shows that end after a dramatic and tense climax.
And of course, episode 00 "The Adventures Of Mikuru Asahina", which features the SOS Brigade's movie, complete with satirical narration by Kyon. In the original airing, this was the first episode.
The first half of Serial Experiments Lain episode "Infornography" eschewed narrative altogether and combined a Recap Episode with a non-linear collage of images and moments from previous episodes, at rapid speed.
The manga had two chapters that turned the series into a high school drama, and one that turned the series into a hospital drama.
For one whole episode in Cromartie High School (and its equivalent three-part manga arc) the tale of Gorilla Sushi is presented, in which the Gorilla of Cromartie uses his banana sushi to bring order to a broken father-son relationship in the Sushi restaurant.
Almost all of the final episode was spent on recreating most of the first episode... only with the entire cast genderbent as females.
Higurashi no Naku Koro ni is normally a psychological horror series, with some comedy at times. The first episode of season 3 is entirely comedy, and revolves around the characters attempting to get Keichii's briefs off. Normally serious characters who were shown dying horribly in other episodes do things like having a bullet proof chest and throwing syringes at someone to get their magical swim trunks, to the point where it's more like a well written crack fic than an actual episode of the show.
Every volume of Franken Fran has one chapter of Fran deciding to go to school. These tend to be more light-hearted than the usual fare of horror since the students treat Fran and her experiments as completely normal.
Even moreso "Cockroaches 2", which deals with Veronica's interaction with a civilization of sentient cockroaches, complete with superheroes.
Naruto Shippuuden episode 82 takes the traditionally shounen tone and style of the series and runs over it with a tank. Most people who watch the first thirty seconds alone will begin to wonder if they accidentally found the wrong show, and it only continues from there. However, despite this, fan response to this particular episode is generally positive, due in large part to the fact that said episode focused on Shikamaru and greatly developed his character. And the fact that it was awesome.
The first actual filler arc of Naruto is essentially a breather after three long and tiring arcs. It then gets abused in the last 90+ episodes.
The manga had a chapter that look a break from ninja missions, training sessions, and character drama, and instead had a series of gags centered around Naruto, Sakura, and Sasuke trying to see what Kakashi looks like underneath his mask, ending on a hilarious Anti-Climax. When adapted for the anime, however, an encounter with some enemy ninja was added to pad out the run time and make it not so completely different.
Mahou Sensei Negima! goes from a cliffhanger about Asuna being captured and having a spell canceler targeted on her memory wipe activate while the next chapter introduces the infamousfurogroping scene. After the fanservice quota had been met for the next six years or so the plot returned to training from hell, a not-quite-a-tournament arc (only one important fight) a new new bad guy and a ball.
The manga as a whole can qualify due to the extreme Genre Shift; the first chapters were a Fan Serviceyharem comedy about a young boy going to Japan to teacher a junior high class. Fast forward to the later chapters, which feature this same boy and half the class trapped in a magical world on Mars which is in danger of collapsing, planning to invade the lair of a Big Badwho plans to "save" the inhabitants of the magic world by erasing them from existence.
A special mention goes to episode 5 of the Mou Hitotsu no Sekai OVA, (and to a lesser extent the chapter it's based on) which borders on Hostile Show Takeover: following the intense shonen action, the focus shifts entirely to Yue and her adventures at a Wizarding School, complete with it's own theme song, presenting the whole affair as a Magical Girl/Schoolgirl Series. Especially notable in that it's basically a regression to the genre the series started out as, thowing the massive amounts of Genre Shift into relief.
In the Tsubasa Reservoir Chronicle anime adaption, there was a filler episode in which the main characters (except Mokona) were all stuck inside a storybook world, which Mokona had to write them out of. This is referred to as "the episode on crack" and also features an Art Shift, being that everything in the book is drawn in a cutesy/chibi style.
The Amagai and Muramasa arcs, and the two episode follow-up of the Amagai arc from the Bleach anime may count as they are completely out of continuity when compared to other fillers. Also, every episode involving Don Kanonji, which are even better examples, since they are all canon. Episode 227, which is also canon and, chronologically takes place right before the first episode and the two following filler episodes are also completely out of place.
Episode 7 of Mayoi Neko Overrun! expands the Show Within a ShowYuusha Choujin Grand Braver into a full show, airing its fictional 26th episode. The characters, conveniently enough, mirror Mayoi Neko's cast, but it takes one off guard. Everything except the On the Next gives the impression that it really is a totally different show.
Similarly, episode 20 of To Love-Ru takes its own Show Within a Show, Flame Girl Magical Kyouko and creates an entire episode out of it, complete with its own opening.
Black Butler, for its OVA, performed Shakespeare's famed tragedy, Hamlet. Of course, the the professional theatre troupe couldn't make it, so all the characters have to put on the show. Of course, everyone has their own ideas on their "interpretation" of Hamlet, leading to some truly weird roles, such as the Undertaker as Queen Gertrude, Grell as a very over-enthusiastic Ophelia, and Lau's sister as... seaweed. On the other hand, Ciel and Sebastian made an excellent Hamlet and Laertes (respectively).
Happens occasionally in the Lupin III anime, but this trope really shines in the original manga, with a chapter narrated in Film Noir style, one with the reader as the main character, many No Dialogue Episodes, spotlights on characters that aren't Lupin, ect.