Akahori Gedou Hour Lovege is about two wannabe comediennes who take part-time jobs as superheroines. Due to them destroying the city every time they fight evil, they get mistaken for evil creatures and become feared by everyone.
Akikan! is a series about about empty soda/juice cans doing battle to determine whether steel or aluminum cans are superior, for the standardization of cans into one format, strengthening the industry. Oh, and the cans take the forms of cute girls in strange outfits...
Assassination Classroom is about a class of ordinary junior high students and their teacher "Koro-sensei", a yellow cephalopod-like creature that moves at Mach 20 speeds, which would be strange enough on its own. What pushes it into the surreal is that the students have to kill their teacher within a year before he causes an Earth-Shattering Kaboom... yeah. Not only are attempts on Koro-sensei's life an ultra-common occurrence in the classroom, but the teacher himself happily gives pointers in the art of assassination to his students, even though he is their target.
For a Slice of Life comedy yonkoma, Azumanga Daioh has a quirky and sometimes very odd sense of humor, with scenes like Sakaki's fantasy sequences about a cat-like creature claiming to be Chiyo's father, or pretty much any conversation involving Osaka. To a lesser extend, this applies to the author's follow-up series, Yotsuba&!, a comic about a single father and his very imaginative daughter Yotsuba.
While Bakemonogatari is probably still bizarre in Japan, the religion/mythology of Japan at least let the original audience understand things like the crab gods and lost cows. And the puns. Oh god, the puns...
Bananya: Adorable as all get out, but still widgety. Cats that live in banana peels? Really?
Black★Rock Shooter: Let's get this straight. We have the story starting in a Mental World with a Stripperific outfitted girl fighting other dream versions of girls, no clear heroes or villains, just some girls fighting each other for some reason, and this is before we meet the YandereWheelchair Woobie, Kagari, whose favorite hobby seems to be chucking gross-colored macaroons at people in both the real and Mental World. And an older woman who serves everyone coffee, but calls it "dirty water" when asked if she likes it. Then there's some Fractured Fairy Tale about a bird soaking up rainbow colors until it turns black and dies. And the color of the bird the hero most admires is actually the black dead one.
Bobobo-bo Bo-bobo: the characters confuse their enemies into submission, confusing the viewer in the process. The main character is a Kenshiro knock-off with a blonde afro who uses "fist of the nose hair", and the enemy is an evil empire that wants to shave everyone's head. That covers the first couple of episodes, and it gets weirder.
Butt Detective is a children's anime about a talking butt that solves crimes. Many foreigners found the concept of the show weird, with some of them comparing it to the mid-2000's output of [adult swim].
Cat Soup, which is considered to be weird even by the standards of weird Japanese things.
Chintsubu is a manga about boys who have talking penises. This one is so bizarre that it's often brought up in internet conversations purely as a benchmark of "how weird can Japanese media get."
Cromartie High School is a parody of old shounen shows, about a normal(ish) guy that starts going to school full of "badasses". And a gorilla. And a robot (that doesn't realize he's a robot). And a mute man that looks strangely like the deceased lead singer of Queen.
In Date A Live, there are girls who can cause massive damage to the world just by appearing. And killing them is just about impossible. The solution? Date them. And the Dating Sim-like choices that occasionally pop up as you do so are not decided by you, but by a group dedicated to this very purpose. An idea that only Japan can come up with.
Excel Saga. The American release even has the title written in faux-Japanese letters.
From the same people: Puni Puni Poemi. Two OVA episodes of sheer insanity. The 'magical girl transformation' involved shoving a knife up the rectum of a talking dead fish.
Fighting Foodons, which gives a whole new meaning to the word "food fight". More specifically, due to magic cards called "Meal Tickets", food can be brought to life, and so chefs are constantly competing to create the strongest food warriors.
FLCL. It's pronounced 'fooly cooly' (not even the characters know the meaning), and is about a crazy woman who beats a young boy with a bass guitar to summon giant mecha out of his head so she can kill them, in between hitting him with her Vespa just for fun.
Gintama. Its humor relies on a lot of Japanese puns, references to Japanese pop culture, Japanese-style humor, and a basic knowledge of famous historical Japanese figures. Though later on they also have gags like "Willis Smith" and sneezing the name "Mai-ke-ru Jackuson!" And a whole lot of in-universe running gags and random Widget Series staples. And it has two Star Warsparodies.
Gloom Party takes the cake. It's a yonkoma series that the English publishers, DMP, knew would be incomprehensible to an American audience, since a large amount of the gags are Japanese puns, or refer to Japanese phenomena. Therefore, they added the words How to "read" manga to the title, made sure that the American edition contained only the strips that Western readers wouldn't understand, and added a short explanation of the joke to every strip, turning it into a guide to incomprehensible Japanese humor.
Gonna Be the Twin-Tail!!: Extra-dimensional invaders attack Earth not for its resources or to conquer it but... for its twintails? Rather, they're attacking Earth to drain the "zokusei" (something that has no direct definition in the West) of its people. Said Zokusei manifests in typical Otaku fetishes like bloomers, model-making, and of course, twintails.
Goodnight Punpun, in which a mute little bird (think Woodstock but three feet tall), drawn in line style while everyone else is drawn realistically, tries to comprehend his bad home life and the behaviour of people at his kindergarten. Occasionally, he summons God for answers; God has an afro and is getting tired of the same questions.
The Gothic World Of Nyanpire: An abandoned black cat is found dying alone during a rainy night. The same night, a vampire shows up and feels bad for the kitten. He decides to cut his finger and feed a drip of his own blood to the cat. Which results in the cat turning into an actual living vampire cat. He gains fangs, wings, and a yellow cross on his stomach, and later gets the name "Nyanpire" since he's a vampire. He lives with an owner who owns a Siamese cat named Chachamaru. He later befriends a Samurai cat named Masamunya who starts developing a crush on him. And a fallen angel from heaven named Nyatenshi who is constantly seen chasing a cat named Katsuo by holding a stick over a sardine that Katsuo has.
Gugure! Kokkuri-san: A self-proclaimed doll girl named Kohina decides to summon the fox spirit on her own and ends up being haunted by the Kokkuri...who is so appalled by her low living standards that he appoints himself to be her guardian. He is later joined by a masochistic Inugami, who is obsessively in love with Kohina and an alcoholic bum of a Tanuki who freeloads in the Ichimatsu home to create more weird hijinks in Kohina's life.
Haré + Guu: A boy and his mother live in the jungle, until the mom adopts a Humanoid Abomination in the form of a little girl who has another world in her stomach. Weirdness ensues.
Hayate the Combat Butler, a series about a boy whose parents stick him with an enormous debt to the Yakuza, so he tries to abduct a little girl and ends up as her butler. The series also features aliens, robots, ghosts, demons, talking animals and involuntary time travel.
Hentai Kamen. It's about a martial artist slipping a pair of panties on his head (by accident) and transforming into... a guy with underwear on his head.
Hetalia: Axis Powers (and Hetalia: World Series): World history + countries turned into impossibly cute/hot guys + Ho Yay = this show (especially the English dub, in which most, if not all, of the dialogue is more risque than what the Japanese version has).
Heybot! ostensibly parodies Merchandise-Drivencompetition series while being the weirdest one ever, as the premise involves a kid and his screw-themed robot partner partaking in joke-telling contests. Random chaotic situations (added emphasis to random) and crude humor ensue.
Jewelpet becomes more bizarre and laden with Japanese ancient and pop culture references starting with its third season, which alienates a lot of non-otakus.
Story wise, JoJo's Bizarre Adventure starts out as a fairly normal take on a zombie/vampire tale, albeit with some definite twists. But then there's the names, which almost all reference western rock and roll (Robert E. O. Speedwagon, Tompeti, Oingo and Boingo, etc.). And the art, which largely consists of muscular men in elaborate clothing striking model-esque poses. And some crazy character designs, such as Jotaro, whose hat seems to morph into hair about halfway around his head. Then in part 2, Battle Tendency, the cyborg Nazi shows up and the craziness just grows exponentially. Part 3, Stardust Crusaders, is both the story arc where things really start to become totally insane, and initially the only part of the manga to be officially released in the US (until the earlier parts got released). America even got the Capcom fighting game based on Stardust Crusaders, which is the origin of the ZA WARUDOmeme.
Joshiraku: Cute girls having quirky conversations about inconsequential subjects in the dressing room of a rakugo theater = rakugo being a uniquely Japanese form of comic storytelling by a single performer sitting still in the middle of the stage with only a fan and a piece of cloth as props.
Kamen no Maid Guy that's Masked Maid Guy in English. A gigantic masked sociopath in a maid uniform terrorizes an absurdly well-endowed samurai schoolgirl for her own good. You just don't get that particular kind of "huh?" anywhere else on Earth.
Kujibiki Unbalance is part parody, part homage to every anime genre Japan has produced. The resulting mix of postmodernism and Clichestorm is strange, especially to viewers who don't know the genres in question.
The Legend of Koizumi. World leaders use extreme high-stakes Mah-Jong to decide everything from deals to papal elections to SAVING THE WORLD! Everything is over the topshounen, playing all of the stereotypes of nations and their leaders entirely straight.
The basic premise of My Bride is a Mermaid is explainable (boy gets saved by mermaid; must marry her to keep up The Masquerade), but the execution of the premise is nothing short of insane, involving Mermaid Yakuza, a Terminator (really), and more Art Shift than one would think could be crammed in.
Nerima Daikon Brothers, from the director of Excel Saga. An anime musical series about farmers who want to become musicians but are constantly low on cash. One of the characters falls in love with a panda. Aliens appear.
The fact that One Piece is so insanely popular in Japan, but rarely becomes more than a Cult Classic even among manga in other countries, is probably one of the things that cement this trope.
Oshiri Kajiri Mushi, literally Butt-Biting Bug. Butt. Biting. Bug. This is an anime based on a children's song about a type of bug that makes people happier and more social... by biting their butts.
Ouran High School Host Club relies heavily on Japanese puns and wordplay, and might be a little bit strange to someone with little or no understanding of either.
Ox Tales. A cartoon, based on the Dutch comic Boes, about a Dutch farmer and his misadventures around his farm, including a lot of surreal Body Horror gags involving farm and exotic animals.
Pani Poni Dash!. It's about a school full of very weird students. And a ten-year-old MIT grad teacher. And a cat who says he's God. And a bunny who only exists to be abused. And space aliens who have little to no impact on the plot except to make Star Trek jokes. And a class rep who defies all logic or sanity. And apparently it's all shot on a soundstage and the cinematographer is very bad at hiding it.
Panty & Stocking with Garterbelt is surreal both plot-wise and design-wise, using Western animation art styles for the most part. Namely in the style of 90's American cartoons that were WHAT series in their own right.
Patalliro! is an old-school example. It's a very weird shoujo comedy with BL elements that started in the late 70s and is still running today.
Penguindrum: Two brothers must find a mystery object known as the 'Penguindrum' for an entity residing in a penguin-shaped hat that is possessing and keeping alive their Ill Girl sister, in company with small cartoonish penguins that only they can see. And then the copious amounts of Mind Screw start.
Potemayo. A boy finds an over-possessive moe-blobin his fridge. And he calmly adopts it, naming it after what he had in the fridge. Then another one appears, except this one is a Tsundere, scythe-wielding moe-blob that Beam Spams from the worms on the side of its head, and leaves gifts of charred, bleeding animal carcasses on the desk of one of the boy's classmates. That's the first episode.
Pui Pui Molcar is a series of shorts about the misadventures of guinea pig cars. Not cars in the shape of guinea pigs, mind: guinea pigs in the shape of cars, that walk on their wheels, have windows built into their skin, and poop out trunk cargo when stressed. Sometimes it just follows a normal day for them, but other times they can be squaring off with zombies or participating in high-octane action movie stunts.
Rumiko Takahashi has created a lot of weird manga, with Urusei Yatsura having things like a giant ghost cat who enjoys having tea parties with a little person monk and a school principal and a Jerk Jock who commutes to school in a giant helicopter-mansion, and Ranma ½ having hot water springs that can turn you into everything from a girl to animals.
Saint Young Men is about Jesus and Buddha sharing an apartment in modern-day Japan.
Sarazanmai has ass-eating kappas, a pair of cops who break out into random homoerotic musical numbers, and the villain is a talking otter. Since much of the series revolves around kappa and the folklore surrounding them, most non-Japanese viewers who aren't familiar with kappa are inevitably going to be very confused.
Seven of Seven: A normal schoolgirl is subjected to Cloning Blues when she's split into seven copies of herself... who then all try to go after the boy she was interested in. Eventually, they take turns being the 'real one'.
And it was created by the guy who directed Giant'. Freakin' 'Robo.
Super Milk Chan. A five-year-old girl lives in a house with her robot maid and complains about all the bills she has to avoid paying to her Camp Gay landlord. She's a superhero but she constantly blows off the President (of Everything), who attracts flies. That's just the beginning of the insanity in this show.
Tuxedo Gin centers around a teenage boxer who is killed by gangsters and reincarnated as a penguin.
Ultimate Teacher is beyond description. The most normal thing about it is that the school is so bad, there's a graveyard for dead teachers. Most of the 50 minute OVA is a giant Big-Lipped Alligator Moment. Zombies? Check. Men wearing girls' gym shorts? Check. Guy who fights by flinging quarters? Check. That's not even getting into the genetic experimentation. It cannot be quantified.
The World God Only Knows. Dating sim otaku is landed with a demonic contract to make girls fall in love with him in order to drive out the evil spirits hiding in their hearts, and decides to do so by using dating sim strategies, which turn out to work astonishingly well. There is no possible way an idea like that could have come from anywhere but Japan, plain and simple.
Yakitate!! Ja-pan is a show/manga about bread. No, it does not teach you how to make the bread (one exception for bread-in-a-rice cooker). It focuses on tournament-style battles between bakers. And puns. Lots and lots of delicious Japanese pans.