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Ao No Exorcist's True Cross Academy is this, due to being also a base for the Knights of the True Cross aka Exorcists and where the Japan Branch's cram school/training academy for budding exorcists is. Also, the Headmaster is a demon more specifically one of the Demon Kings and second eldest Satanic son and our main (and titular) character is the bastardSon of Satan, aka the resident living Weirdness Magnet that amps up the weird to Up to Eleven.
Mahou Sensei Negima!: At Mahora Academy, you need rock climbing gear to traverse the library, the School Festival involves Humongous Mecha and zeppelins, the majority of the staff (from what we've seen) are secretly wizards, the center of campus is the World Tree, and if a student there isn't supernatural, paranormal, superscience, or otherwise out of the ordinary, it probably means they have motivational issues. Even the Muggles are bizarre or unusual in mundane ways- like being the heir of a richer-than-an-entire-country family, being able to do way more with a gymnastics ribbon than one should, etc.
Academic Cities in Chrome Shelled Regios, which are giant walking cities run and populated entirely by students and specialized for education and providing for students' needs, like a giant university town. Combines this with City of Adventure.
Rosario + Vampire. This is the natural consequence of creating a school to teach monsters with poor self control how to maintain the Masquerade. Especially when you trick a human into going there.
Beelzebub: The main setting for the first arc of the series, Ishiyama High, is famous for housing the city's worst delinquents, with a delinquency rate of 120%. There are always fights, no one seems to ever attend actual class, and the entire building is smashed up and covered in graffiti. And this was before Oga "adopted" Baby Beel.
Yu-Gi-Oh! GX: Duel Academy. Doesn't help that it's on an island, and thus almost the only setting where any dueling takes place...
Gakuen Alice. An "Alice" is someone with a strange or paranormal talent. Any kid who is identified as having an Alice is legally required to go to the Alice Academy, and cannot leave the campus until graduating from high school at the age of 20. The campus is immense and includes forests and a town. All teachers and administration are Alices, and other staff/residents include robots or mutant animals created by Alices. The buildings inlude three heavily guarded and mysterious school buildings and residence buildings for students and three buildings belonging the three enigmatic headmasters, all of which are guarded with supernatural means. Most of the adventures in the story are set in and somehow related to the buildings and their peculiarities.
Teacher: But really... compared to the times when we had to teach Izaya-kun and Shizuo-kun, this school is way more peaceful nowadays. Back then we used to have filled gasoline drums rolling down the third floor of the school building...
Every high school in Ikki Tousen qualifies. One wonders how these institutions still attract students, considering all the mayhem that goes on there.
Fuuka Academy in the original. In the anime, the girls' identities as HiME are kept secret from the public. In the manga, the identities and the abilities of the girls are well-known, and all of the battles that take place within the school (sometimes between other students) are seen in full view of everyone, as is the resulting property damage.
Garderobe Academy in Mai-Otome, a training ground for Magical Girl Warriors where potential Otome candidates are expected to show poise, grace, and fighting prowess almost in equal measures.
In Medaka Box, there's ''Sandbox Academy". Teachers are conspicuously absent save for the Principal. And many students have a power level on the tier of avatars of Hindu Mythology. The protagonist is often compared to a god, and she might as well be one.
Exaggerated (likeeverythingelse) in Kill la Kill with Honnouji Academy. Students wear uniforms that turn them into superpowered soldiers, club presidents (from boxing to tennis) have their talents amplified dramatically and weaponized, the committee chairs can turn their uniforms into Powered Armor and tanks, the student council president is an Evil Overlord, battles resulting in explosions that send hundreds of people flying happen daily, and that's just the beginning of it.
Kuoh Academy in High School Dx D attracts all sorts of strange students, starting with two packs of devils and eventually including an angel, a fallen angel, and a Valkyrie, among others. It turns out that as well as being a legitimate place of education for humans, devils use it as a front to give themselves a safe haven during daylight hours and a base of operations at night.
The Lovecraft Academy from Locke & Key. The main cast attend the school, and before them, so did their Dad. Needless to say, the main 'big' mystery of the series revolves around what happened to their father and his friends during their school days.
Spider-Man went to both an Adventure High School and Adventure College, and generally continued to do so in every adaptation of the character. As a matter of fact, several of his most memorable villains and allies were either students (Harry Osborn, the second Green Goblin), introduced through students (for example, Molten Man), or as teachers/professors (for example, The Jackal), but even the everyday goings on of the school have a bit of adventure to them. Later on, he becomes a teacher at a high school, which inevitably becomes an Adventure School as well, even after he resigns.
In particular, his college Empire State University, is a huge Adventure School that has since extended far beyond involving just Peter Parker. Just look at the notable faculty/students list, which involves X-Men, villains of every kind, and the Human Torch! Things are rarely boring there.
The Xavier Institute is an example of both this and Superhero School; its students are constantly involved in adventures and attacks by villains and such because its entire faculty and student body are mutants, and, in order to deal with the constant threat of danger, among other things, there's a jet, the Danger Room, etc and the faculty/senior students double as a Super Hero team on the grounds. The school's been leveled so many times by these threats, it's been lampshaded by the cast.
After the above school is destroyed, Wolverine opens the Jean Grey School for Higher Learning. Its students include a brood, and one of the teachers is the time travelling child of the namesake Jean Grey. Homework is basically training for adventuring, e.g jetpacking and snow ball fights.
Cyclops opens the New Xavier Institute. It is located at the old Weapon X Facility, the place that gave Wolverine his unbreakable skeleton and sharp-ass claws, and the students regularly go out on missions with the teachers. Said teachers consist of: A mutant revolutionary who shoots lasers from his eyes, and is wanted for the murder of Charles Xavier, a mutant telepath who can turn her skin into diamond and wears Stripperific outfits, a mutant Nazi-survivor who can control magnetic fields and a sociopathic mutant sorceress who regularly takes the students through limbo when teleporting and time travels.
In Static Shock, it's justified in that the chemical explosion that caused the "Bang Babies" happened at a popular local gang hangout near Virgil's school.
This is the setting for PS238, which is about an Adventure Elementary School filled with superpowered children. Despite their best efforts, they don't do that great a job of keeping the super powered danger away.
Advanced Public School #001 (APS-001) in the Gold Digger universe. In the "Tifanny & Charlotte" series, the title characters attend the school.
The Imperial Naval Academy in The Odyssey definitely qualifies as this. It's run by an incredibly ancient Dragon-Blooded and designed to prepare the students for all of the threats they might encounter on the high seas. Under the new leadership, it's meritocratic, and death can be a consequence of failure.
Both Furinkan and Hogwarts in Top Dog. At Hogwarts, it's because it's a political football that is politically neutral in all disputes, and members of species and cultures that are furiously at war both attend; at Furinkan, it's more a function of poor faculty judgment in approving projects. (The tokamak in the physics building ended up blowing up, and the biology projects routinely escape and eat people.)
The University of Wisconsin becomes the site of several battles in Power Rangers GPX. The other battles take place off-campus in Madison. Southern Illinois University in Carbondale serves a similar purpose in an AU fic.
There's Mildwood University in The Impairment which we see through the eyes of protagonist Kyle Griffin, a freshman suspect framed for the murder of his roommate at the hands of an extra-terrestrial. Said school features students overwhelmed with fear from a two year long string of similar murders, a fraternity of hostile young men, and blood thirsty aliens which, of course, only Kyle can seem to see and are connected to the one Kyle saw kill his roommate.
Mademoiselle Geraldine's Finishing School for Young Ladies in The Finishing School Series. The girls are taught espionage and assassination alongside manners and graces. It is located in an airship and the instructors number one vampire and one werewolf respectively.
Percy Jackson and the Olympians: Camp Half-Blood is a summer camp designed to protect the children of Greek Gods, and teach them battle skills - but if that doesn't count as a school, then every school Percy's been to counts, because monsters just can't get enough of his godly goodness.
The titular academy in the H.I.V.E. Series is a school for supervillains. In the first book the school is attacked by a giant plant - a science experiment Gone Horribly Wrong.
The Gallagher Girls: Gallagher Academy for Exceptional Young Women is a school for training teenage girls to be spies.
The Seven Realms gives us Oden's Ford, a world-famous academy that teaches everything from magic, to the art of war, to how to be a proper lady. Most of book two takes place there. The Imagery is spectacular.
Brooklyn House and the larger House of Life in The Kane Chronicles count. You'd be hard pressed to not find magical activity at either.
Divine Blood Novels: Bravura Academy is more or less a normal boarding school that boasts both a college and high school campus. The high school level kids are already learning college level courses. While the school itself is focused on normal student hijinks it is the focus of several power groups. Bravura itself is sponsored by Avalon who also uses a front company security organization to test potential recruits. Then there's the students, which include an Avalon soldier, a "Visionary", a Manic Pixie Dream Girl, three demi-gods, a Demoness, and a werewolf.
In the Star Wars Expanded Universe, the various Jedi and Sith academies all have shades of this. It is especailly notable in the Jedi Academy trilogy in which a Sith Lord appears.
In The Unexpected Enlightenment of Rachel Griffin, Roanoke Academy for Sorcerous Arts. Day three has the heroine musing that she's made friends, gone flying, discovered a world-spanning conspiracy, prevented a murder, and already been to the infirmary twice (once for ending on the wrong side of a Baleful Polymorph spell, once to rescue a classmate being drained by an wraith), and "hoping" the rest of the year won't be so eventful.
Xander: Something weird is going on. Isn't that our school motto?
Community: Greendale Community College has all of the madcap, wacky adventures you could ever hope to find at any college.
Beacon Hills High School in Teen Wolf. It has five werewolves currently attending it, a werewolf hunter as its (former) principal, and at least a few teachers who are aware of or involved with the supernatural themselves. A lot of weird stuff happens there.
Maritsu Evil Academy from Disgaea 3: Absence of Justice. The school's campus is so gigantic that it encompasses most of the world (And presumably will be the world eventually, due to its constant expansion). As such, traversing things like icy wastelands and lava-filled ruins to get to a class isn't out of the ordinary, making the "adventure" part of the trope's name quite literal in this case.
Timmy Turner's Dimmsdale Elementary not only always has the effects of Timmy's wishes to deal with, but the mad fairy obsessed Mr. Crocker to make things crazy as well.
Codename: Kids Next Door features a school where some of the students are in a global organization bent on guaranteeing freedom for children worldwide, teachers who oppose this group, corrupt members of the student council who try to ensure that snow days never occur, playgrounds with hidden paths to volcanoes, etc.
The secret temple that leads to the Fountain of Youth, kindergarteners that are practically wild animals, Wild West situations... the list goes on and on.
Hamilton Hill High School is this in Batman Beyond. Aside from Terry "Batman" McGinnis and Max "Reckless Sidekick" Gibson, this school has a Flash Thompson-esqueJerk Jock, a Lovable Alpha Bitch who seems to attract trouble, a student who dated a sexy psycho robot, a student whose step-father had a short career as a villain, a student whose thought to have been dead birth father tried to kill her adopted father, an avid gamer who almost helps a villain kill someone, no less than four students who are villains, a coach who attacked Batman trying to protect students using steroids and a guidance counselor who was a super villain.
Casper High in Danny Phantom qualifies. It's a wonder the teachers could get any teaching done with ghosts constantly fighting in it.
All three schools in Winx Club: Prof. Avalon at Alfea turned out to be an imposter cloud spirit designed to kidnap Bloom; Cloudtower turned out to be housing three delinquent teenage witches who never seemed to go to class (and the school was the usual occupying hideout for the seasons evil villain); Redfountain is a school for young heroes and the headmaster is a wizard, but none of the boys have magic.
While nothing too out-of-the-ordinary happens, Third Street School usually has something crazy going on every episode.
Ultimate Spider-Man's Midtown High is where Spider-man and four other hero's attend high school, the school is constantly attacked by dozens of super villains and protected by SHIELD's security system, and its principal is an undercover SHIELD agent.
Justified in Code Lyoko with the Kadic College: XANA targets almost exclusively said college because the Lyoko-Warriors usually are here. Ironically, none of the events that happen remain in the memory because the Lyoko-Warriors possess their own personnal Reset Button.
Detentionaire's A. Nigma Hish School. Let's see: a weird snake/dragon/lizard monster thing called a Tazelwurm (who also happens to be the school's mascot) crawling around in the vents, secret passageways literally everywhere, an eye-patched cyborg principal, a mysterious pyramid under the school, most of the teachers being clones of famous people, vacuum-wielding hazmat janitors, mysteries, conspiracies... there's quite a lot of adventure to be had.