ものの怪 Mononoke (not Princess Mononoke)
What it's about: A quiet medicine seller, who always insists that he is just a medicine seller, wanders throughout Japan and its eras exorcising the demonic presences known as mononoke.
Why it's good: Mononoke is a collection of all the things that are interesting about Japanese art and culture and remixes them into a horrific pastiche that is undeniably cool. And the medicine seller is like the Doctor Who of Japan.
Best part: I'm rather partial to the incense episodes, wherein they play a crazy game that actually exists in real life but seems like it never could.
瀬戸の花嫁 Seto no Hanayome (the Inland Sea Bride)
What it's about: A young boy's life is saved by a yakuza mermaid. Rather than have him be killed for knowing her secret, she chivalrously marries him (chivalry is connected to yakuza in Japan, so you can guess what kind of world the mermaids have). Of course, her father, the leader of the gang, is very unhappy and tries to kill his new son in law constantly.
Why it's good: Where to start? We have funny yakuza language, Saru's monkey antics, Shark's lilting polite language versus his blood lust, Runa's terminator father, the song that plays when things get chivalrous... The whole show is a collection of odd jokes.
The best part: Probably, and I say this realizing it will take a second for you to digest it, the way that Nagasumi-kun and his mother are in love with afro-dandy Masa-san.
十兵衛ちゃん Jubei-chan series 1&2
What it's about: The Jubie series are the tale of a young girl that discovers she is heir to the lovely heart-shaped eye-patch of the deadliest swordsman in Japanese history. The first series is more comedic and unrefined, while the second is polished and cool.
Why it's good: Each series is good in it's own way. The first shines in terms of comedy, while the second takes things to a new level. Suddenly the sword fights are superhuman, and the rule of cool takes over.
The best part: For me, the star of both series is the ruffians. I couldn't find any good videos of them, so you will have to settle for this clip of the epic sword fighting.
セクシーコマンドー外伝 すごいよ!!マサルさん Sexy commando
What it's about: Sexy Commando is actually a series by the same director as Jubei Chan, but it came first and is based on a gag manga, so it is much more crude and experimental.
Why it's good: Sexy Commando takes all the rules and conventions and ignores them to the point where it doesn't even have a proper ending. But that's the joke. It unfortunately had a lot of repeated songs as filler for the super short episodes, but otherwise, it is one of the best gag animes out there. The sexy commando of the title is a fighting style that depends on weirding out your opponent to create openings.
The best part: Definitely the principal of the school. He's an old man that speaks with an odd accent and is secretly a sexy commando. Also, he dresses as the student Susan. In this clip, he suspends the students to keep them in the sexy commando club.
さよなら 絶望先生, Zestubou Sensei 1&2
What it's about: Mr. Shiki's name is twisted via kanji word-play into despair, and he has a personality to match. He takes a negative view on all things connected to Japanese life. His students all love him, in harem-anime tradition, and they all embody odd Japanese stereotypes.
Why it's good: Satire is something that is usually lacking in Japan, so all the desperation is surprisingly refreshing.
The best part: I'm rather partial to Kaere Kimura, a student that was abroad too long and now thinks Japan is an odd country, while being myopic about her own foreign foibles and tempestuous nature. I totally identify with that.
ギャグマンガ日和 Gyagu manga biyori (A Good Day for Gag Comics)
What it's about: The story is different each time, but the series is basically an exploration of how much inanity you can pack into five minutes of airtime.
Why it's good: It's unlike any anime you've ever seen, and provides lots of interesting jokes that lend to repeat viewings. Also, it's excellent to share with company in a Golden Eggs sort of way.
The best part: Harris DESU!
フリクリ Fooly cooly/FLFC
What it's about: On the surface it is just a cool show about a boy who's head is a dimensional door for robots and other horrors to pop out of. But it's really about coming of age. Leave no Freudian stone unturned, viewers.
Why it's good: Stylish and random, Fooly Cooly stays cool all the way to the end of it's short run without losing steam.
The best part: I couldn't choose. Which is just as well; all the YouTube clips are fan made music videos, which are all a plague upon mankind save one.
妄想代理人 Paranoia Agent
What it's about: Imagine what would happen if our collective-unconscious gave life to an urban legend that likes to smack people with a baseball bat. Then imagine if that entity started to grown exponentially in power.
Why it's good: This is the brainchild of Satoshi Kon, the guy that brought us the feature-length animes Tokyo Godfathers and Paprika. In Paranoia Agent he transfers his expertise to the small screen and provides a truly unique tale that really evokes Japan well. I first saw some of this series dubbed in English and didn't get it, but after coming to Japan and hearing the original voices, it is much better.
The best part: The opening itself. As is appropriate for this show, it creeps you out and the lyrics are haunting as well.
I could mention a few more (yes, yes, everyone loves Azumanga Daiō), but good grief this is a huge enough post! If you have something really interesting to recommend, shoot me a comment, Dear Reader.