So here's a list, categorised by loose genre, covering things I've enjoyed. The vast majority are available on crunchyroll, and links are included. A few aren't, and so no link is included.
- Neon Genesis Evangelion One of the all-time classics, of course, and a personal favourite. It's dark and depressing, it's extremely psychologically complex, and it has both strong religious symbolism (Gnostic Christian) and some strong religious themes (Buddhist). One of the original genre-defining Mecha deconstructions, and likely to remain popular. HIGHLY RECOMENDED.
- Bodacious Space Pirates is the story of Kato Marika, a teenaged girl who inherits a letter of marque from her deadbeat father and so is now the legal heir to a line of privateer captains going back over a hundred years. Lots of great world building, lots of great character work, lots of clever plotting, and loads of strong central independent female characters. Relatively little fanservice for the genre, and it keeps the balance of serious to tongue-in-cheek properly maintained (to my mind). HIGHLY RECOMMENDED.
- Campione! is one of my personal favourites, though the series is not as good as the novels it's adapted from. It's about a Japanese highschool boy who hunts and kills ancient gods using a set of powers which require him to know the god's mythology and history in great detail in order to use them, information he acquires via ritual magical knowledge transfer whilst making out with one of his four beautiful girlfriends. And described like that it sounds like the most absurd fanservicey rubbish around ... but it isn't. It's actually got strong and well written characters, clever use of obscure mythology, a well tempered and wry humorous tone, and a surprisingly nuanced view on the difficulties of a polyamorous relationship. HIGHLY RECOMMENDED.
- Fate/Stay Night: Unlimited Blade Works is the adaptation of one of the routes from the famous Visual Novel Fate/Stay Night. It features young, relatively idealistic mages dealing with the darkness and horror of an all-out mage war, whilst trying to make a difference and improve things. Also the best Tsundere ever. HIGHLY RECOMMENDED.
- Fate Zero is the prequel to Fate/Stay Night, though they can easily be watched in any order. It is a dark, violent, brutal take on a series of horrible horrible people fighting a magical war in a small city in Japan. Far and away the darkest entry in the Fate franchise, it's very enjoyable if you are into that kind of darkness. HIGHLY RECOMMENDED (WITH CAVEAT).
- Asura Cryin' is a pretty fun series of magical mecha battles and rival school councils. Kind of loses the plot a bit in the second half.
- Blast of Tempest Two shakespeare quoting magnificent bastards take on the entire magical world by themselves despite having no magical talents of their own ... and win.
- Sword Art Online of all the many "trapped in an MMO" series this one is probably the most popular. The first half of season 1 is essentially a fantasy sword hero story combined with a strong romance subplot with a "trapped in an MMO" gimmick as the establishing element. The second half is horrible in all kinds of ways which would really deserve a whole post to explain by themselves. The first half of season 2 is fairly generic but ok, even if the main character is annoying in his tendency to deprotagonise his female supporting cast. The third quarter of season 2 is a pretty generic fantasy quest story. The final quarter of season 2, on the other hand, is a great piece of character driven SF looking at the impact of VR technology on unexpected parts of people's lives. Over all it's hard to know where to place the series because of the highly variable quality.
- Pale Cocoon the very best conceptual SF short film I've seen in a very long time. Only 25 minutes long. WATCH IT.
- Crest of the Stars/Banner of the Stars (I, II, and III) is a slow-burn romantic space-opera set against the background of a galactic war. It features an extremely well developed and complex constructed culture, complete with its own conlang. The series is quite happy to take time just slowly dealing with social stuff rather than having to keep up with action -- though there is action. HIGHLY RECOMENDED.
- Last Exile Diesel-punk drama dealing with the politics and warfare of a very strange society told from the perspective of a pair of young delivery pilots who get sucked into a major international conflict. HIGHLY RECOMENDED.
- Shin Sekai Yori (From the New World) is a piece of atmospheric post-historical science-fiction set in a transformed world of disturbing psionic powers, alien social structures, and inter-species oppression and politics. It's really very good, but pretty grim. HIGHLY RECOMMENDED
- Time of Eve Asimovian Robots used to examine questions about personality and identity. HIGHLY RECOMMENDED.
- Gargantia on the Verdurous Planet In the far future a soldier from an omnicidal space war gets thrown through space to a planet previously only known of in myth: Earth. On Earth he learns to live, breathe, and eat freely, and possibly even to love, whilst we learn about just how dystopian his space-society is, and what the secret is which caused the start of the great war. HIGHLY RECOMMENDED.
- Yona of the Dawn is the tale of a once-pampered princess who is forced to wander the land in medieval pseudo-Korea with an assortment of bishie protectors, and in the process we get a lot of social commentary, politics, and not a little bit of romantic comedy. HIGHLY RECOMMENDED.
- Log Horizon is another "trapped in an MMO" series, but this is the one which uses it as an excuse to get into heavy world building, economics, and political discussions whilst also engaging heavily with the nature of the reality the characters are in and the fact that a world that works according to MMO rules has some very exploitable loopholes in it. HIGHLY RECOMMENDED.
- Humanity Has Declined A surreal black comedy involving the last generation of humans in a world now populated by incomprehensible post-human creatures.
- Kamisama Dolls Wooden low-tech mecha, small village politics, very twisted and visceral violence (including references to sexual violence) and some extremely unpleasant people add up to a very dark take on hidden powers within the modern world in which those people with the power make use of it to abuse and belittle those who lack it.
- Dusk Maiden of Amnesia is a slightly weird mixture of ghost story (and really very creepy one in places) and romantic comedy hijinks (well done, for the most part) about a ghost girl who haunts a room in a school and the psychically sensitive boy who runs the supernatural activity club in that room. The creepy as hell anime-specific ending is actually toned down compared to the ending of the manga. HIGHLY RECOMMENDED.
- Mysterious Girlfriend X adolescent sexuality and coming to terms with your developing attraction to a member of the opposite sex told through the medium of saliva. No, seriously. It's really good and well made and stylish. But it's squicky as hell for a lot of people. HIGHLY RECOMMENDED (WITH CAVEAT)
- Otome Yokai Zakuro during the westernisation of Japan a group of young male single military officers are paired up with young female single half-spirits in order to help police the spirit world and keep them in line. Romance ensues ;) HIGHLY RECOMMENDED
- Kotoura-San is the tale of a girl born with the power to read minds (and no power to not do so), whose life has been horribly messed up by the results of this psychic affliction which has destroyed all her friendships and familial relationships ... and then she meets a boy who is simple, straightforward, and really really nice and slowly her life starts to change for the better. HIGHLY RECOMMENDED.
- Say "I love you" a romantic drama in which a troubled isolated girl with trust issues falls into a romantic relationship with a friendly, popular, well liked boy who is far too accommodating for everyone and has a secret problem with self-esteem.
- Fortune Arterial is a Vampire highschool rom-com with a male human protagonist and a female vampire love-interest. It's pretty decent, pretty amusing in its own way, and I rather enjoyed it, but it's very much a romance, if that's your thing.
- Happiness! is what you turn to when you want an extremely generic rom-com with magic.
- Myself; Yourself fairly standard, but well written, romantic story about a boy returning to his home town after years away and failing to recognise his old girlfriend immediately.
- Chi's Sweet Home is a series of shorts cataloging the adventures of a tiny and adorable kitten. HIGHLY RECOMMENDED.
- Dog Days is an enjoyable romp of silliness in which a highly talented kid who's main passion is running assault courses (which he is a national-level junior champion at) gets sucked into a fantasy world which solves all its wars via the medium of assault courses. It's extremely silly.
- GIRLS und PANZER is an epically silly cute girls being cute whilst doing sports anime in which the sport in question is riding around in tanks. HIGHLY RECOMMENDED
- Hayate no Gotoku ! The world's unluckiest hyper-competent highschool student ends up massively in debt to the Yakuza, but works as a butler for a shut-in ultra-rich little-girl to pay it off. Hilarity ensues (and it really does). Extremely tongue-in-cheek comedy, with a strong element of fourth-wall-breaking meta-humour. Also delightful send-ups of clichéd romantic plots. HIGHLY RECOMMENDED.
- My Ordinary Life Weird slice of strange whimsical comedy in which odd things happen for no real reason and nothing makes much sense, but who cares, because it's funny. HIGHLY RECOMMENDED.
- Free! - Iwatobi Swim Club is a tale of attractive young men with lots of loving animated muscles who are massively into swimming and homoerotic subtext, but mostly swimming. They're presented primarily as eye-candy for the female viewership (represented conveniently by one bland female student) but also as a source of humour as they engage in the kind of silly antics cute girls normally engage in in anime. HIGHLY RECOMMENDED
- Monthly Girls' Nozaki-kun is about a girl named Sakura, who has a crush on a boy named Nozaki ... and it turns out he's actually a successful semi-professional shoujo manga writer/illustrator! Which is a good springboard for lots of humour poking fun at the conventions of shoujo manga. HIGHLY RECOMMENDED.
- Skip Beat! As motivations for getting into show business go "revenge" is not a common one, but it can be very effective. Kyouko is trying to succeed not for her own sake, but to get revenge on her horrible ex-boyfriend. That said, her adventures in the world of acting and modelling are lots of fun, and the characters are very well rounded. HIGHLY RECOMMENDED
- My Little Monster romantic comedy in which a neurotic and misanthropic girl whose life revolves around nothing but studying falls accidentally into that weird ambiguous sort of not-quite-romantic relationship you can get into as a confused teenager with a high-functioning sociopath. HIGHLY RECOMMENDED.
- Tsuritama fabulous highschool boys (one of whom is an alien) bond through fishing. Mostly it's about the fishing.
- Venus to Mamoru! A young spiky-haired boy goes to wizard school and immediately becomes the object of affections for the most powerful and terrifying young witch in the world. The student council (who are also a shady magical black-ops group) think it would be hilarious to play around with this relationship. Basically a very silly rom-com.
- DENKI-GAI is the humorous and romantic misadventures of the staff of a geeky bookstore. They mostly have silly misadventures, and occasionally talk about erotic manga.
- Polar Bear Cafe A Polar Bear runs a café. He likes making puns.
- Poyopoyo the life and times of a spherical cat and the family that has adopted him.
- Oreshura Also known as "My Girlfriend and My Old Childhood Friend Fight Too Much" is a fairly fun piece focusing on a put-upon male protagonist who is caught in the ambit of a manipulative girl who blackmails him into pretending to be her boyfriend.
- Kokoro Connect five highschool students suddenly start swapping bodies without an apparent reason. The series acknowledges how creepy and traumatic this can be, whilst also avoiding playing the angst up too much. Those affected are clearly under psychological stress from what is happening to them, but are also not about to let it control their lives completely. In the end it's a strong story about friendship and unity in the face of externally applied difficulties, as well as about not judging others hypocritically when your own private life contains very similar issues. HIGHLY RECOMMENDED.
- Gingitsune is about a teenaged girl who is a shrine maiden and has the power to see the guardian spirit of her shrine, a silver haired fox named Gin. There's some supernatural stuff (obviously), but mostly it's a pleasant story about her making friends and being a normal girl whilst also having a close relationship with the gods. HIGHLY RECOMMENDED
- Little Witch Academia Sadly only one episode yet exists of this delightfully whimsicle tale of young girls at a boarding school for trainee witches. Feels a lot like The Worst Witch, if you've ever read those books. HIGHLY RECOMMENDED.
- Occult Academy is a story about a highschool-aged girl continuing her Father's work in investigating the occult, paranormal, and weird, and also a time-travelling spoonbender from the future. It's kind of, but the pacing is terrible.
- Soranowoto cute girls do cute things whilst in uniform, acting as the garrison for a sleepy little fortress right on the border in a weird post-post-apocalyptic setting.
- Chihayafuru is a series about a teenaged girl named Chihaya who is completely obsessed with a traditional Japanese card game called Karuto. It's essentially a sports anime cataloging the trials of her and her friends as they work their way through the Karuto leagues and try to compete with the experts in this rather eccentric and old-fashioned game.
- Wandering Son A romantic drama featuring a trans-girl and a trans-boy dealing with the difficulties of transexual identity in modern Japan.
- Space Brothers follows a man who is trying to become an astronaut in the near future, following in the footsteps of his younger brother.