Adolf: Spanning approximately half a century, from the Berlin Olympics to the then-present of The Eighties, focusing primarily on the years leading up to World War Two, it is by turns a Coming-of-Age Story, Spy Fiction, Romance and a fairly well-done Author Tract about the evils of war and racism.
Again!! is a school sports club comedy drama, with time-traveling.
Its Spiritual SuccessorDurarara!! takes Genre Busting even further by featuring students, gangs, mythology, mad scientists, Japanese horror and Anonymous, just to name a few. It seems on the surface to primarily be a sort of mystery/thriller with a bit of action thrown in, but it's also got a not-insignificant amount of slice-of-life often by way of... interesting characters (like an Irish faerie who lives in a high-class Tokyo apartment), but in a way the best single descriptor for it might actually be a romance deconstruction. With some reconstruction later on. And not in any way that one would normally think of a romance being.
Bakemonogatari is a comed- well, a bit, maybe, but it's mainly an action... Hang on, there's not much of that either. Okay, so it's a harem... no, not quite. Romance...? Well, the main character has a love interest, but that's not the focus of the show. If there were a genre for it, it would have to be called "Dialogue, Mind Screw, and SHAFT". The entire show is based in folklore and ghost stories, too.
Cat Planet Cuties About half fairly standard ecchi romance/comedy. The other half is a thriller.
CLANNAD: Word of God states that the main theme is family. However, as the plot progresses, we see slice of life and romance moving into the story, not to mention that Tomoya's Unwanted Harem become True Companions. Fuko's arc borders on fantasy, but the second season takes the cake and eats it with Ushio turning out to be the Girl in the Illusionary World, Tomoya being the Garbage Doll, thus firmly establishing the fantasy aspect of the show. Also, Light Orbs. Need I say more? Mind screw...for the viewer at least?
Also, according to the Alan Moore quote that currently heads this page, Cowboy Bebop is life.
Shinichiro Watanabe: The work, which becomes new genre itself, will be called ... Cowboy Bebop.
Its spiritual sequel Samurai Champloo is set in feudal Japan and melds samurai action with elements of humorous Slice of Life and adventure, along with bizarre anachronisms like graffiti artists, breakdancing, rap, and baseball.
Detective Conan: Pretty much every genre except for fantasy has been covered during the manga's run. There's been everything from romance to Body Horror to wrestling. The Spin-OffMagic Kaito, which is canonically in the same world, actually does have magic. Both real and fake magic, so the world as a whole does have fantasy elements.
The only thing you can really pin down about Excel Saga is that it's a comedy, since the series switches genres every episode. Even the comedy part isn't completely consistent, since one episode is played completely straight (but then again, that's the joke.)
The anime and manga are also inconsistent with each other on what kind of comedy it is. The anime is more of a parody of...everything (including itself,) while the manga starts out as a commentary on post-recession Japan with Slice of Life elements, and then slowly starts introducing more sci-fi and deconstructive elements, and a more serious story.
Figure 17 Tsubasa & Hikaru Slice of Life, Magical Girl, Sci-fi, Horror (dude gets impaled in the second episode, framed alongside such stories like performing a school play written by one of the children). Contains a lot of drama that would alienate children of the protagonist's age yet still deals, very realistically (despite the aliens), with issues a child could struggle with.
Harem Royale - When the Game Ends - is a supernatural high school romantic comedy that includes horror (the romantic comedy is enforced by a literal demon, and the girls who don't win the male lead's heart will be sent to hell).
Haruhi Suzumiya is a little bit of sci-fi, comedy, mystery, romance, and slice of life in a highschool setting.
Hell Girl. A fusion of suspense, drama and horror, with some slice of life and social commentary about the least appealing aspects of the Japanese society thrown in for good measure. The third season is full of Mind Screw as well.
While Hunter × Hunter is primarily a shonen action series, (albeit with massive Deconstructor Fleet overtones,) there are plenty of non-shonen action things going on, like when Gon and Killua spent an entire arc auctioneering, or when they went to Greed Island and the series suddenly became an MMORPG.
JoJo's Bizarre Adventure really likes to hop from one genre to another with every arc, going from globe-trotting adventure story to detective story, to gangster story, to prison story, to racing story. Even the fight scenes involving Stands often get more cerebral than your typical action scenes, and feel more like mysteries than straight up fights.
Kaiju Girl Caramelise is a fairly typical shojo romance story, except that the girl occasionally turns into a Kaiju and prompts the same kind of action seen in Kaiju movies.
Magical Project S started as a parody of magical girls, but then becomes an indecisive parody that sometimes took its elements seriously. Then it explored psychological and deep themes, then explored the concept of friendship. Afterward it went to romance.
Planetes is hard sci-fi, with plenty of drama, strong themes on relationship building, and a few comedy bits to relax the audience. Some people just agreed upon it to be a slice-of-life story, but In Space!
Shakugan no Shana: Is it martial-arts fantasy? Or horror? Or school drama? Or romance? One episode, they are having an epic sword fight with a giant bear and using fire powers. The very next, they are relaxing by the pool outside school, and having a rivalry of another sort.
Have fun listing Slayers as a single genre of anime. Set in a fantasy world, it includes comedy, action, and large amounts of drama...often into a single episode!
Yuusha Gojo Kumiai Kouryuugata Keijiban is a multi-dimensional message board for all kinds of heroes, with each chapter depicting a different thread posted by a hero seeking help or sharing their experiences. Not only is the manga itself very unique, but it also brings up unique types of heroes in-universe. For example, one hero is a king who managed to fight off corruption in his kingdom; another hero is an Earthling who can't decide if he wants to leave his job to jump into the magic summoning circle for the 24th time.
There's an entire genre where every notable show and game has done this: Mons. The Trope Codifier and Trope Namer shows, Pokémon and Digimon, are so fundamentally different that every trope in the genre has been subverted in one incarnation of either series.
Mons are analogous to animals: They are manmade in Digimon, and we have seen real animals in that series.
Your job is to collect mons: Not in Digimon. Okay, Maybe in some of the games, but none of the animes has that.