Last week I wrote about Kino's Journey, one of the few animes I'd recommend to people who don't care about anime. The reason I think that's a useful statement is because I want to draw a distinction between "anime" as a medium of Japanese animation, and "anime" the collection of tropes, stock characters, and, well, bullshit that is generally associated with the form.
If Kino's Journey is number one on this, here's numbers 2-5.
First, a preface to explain my criteria further. There is a lot of bullshit in a lot of anime. Anime bullshit, I call it. I have built up my tolerance to it over years of watching shows, such that I can put up with it and enjoy certain programmes in spite of it. A good example of this might be Read Or Die OVA, a joyfully silly action series in which a half-Japanese, half-English 'papermaster' - someone with the power to bend and shape paper with their mind - joins the British Library Special Operations Division in order to prevent someone from playing a symphony written by Beethoven which causes people to kill themselves from space so that the entire population of the planet can hear it.
It's fun! It has wonderful action scenes and it's about the power of books. But I probably wouldn't recommend it to you if you don't care about anime, because I think you might struggle to tolerate the bullshit parts of it, which includes but is not limited to a character dressed like this. Other reasons I might avoid recommending shows I otherwise enjoy include stock characters, awkward sexualisation of young characters, comedy based around sexism, screaming or ultra-violence, a reliance on a prior understanding genre tropes, and more. Basically: if it's a key feature of Sword Art Online, then it rules out a series.
Which isn't to say that any of the following shows are perfect. They won't be to everyone's taste and they might include small examples of the above, but they're animes which can be enjoyed comfortably, I feel, by anyone. No need to have watched other animes. No need to explain away jokes. No need to say, "Yeah, but it's Japan." as if that means anything.
Satoshi Kon is best known for his animated films - Perfect Blue, Millennium Actress, Tokyo Godfathers and Paprika - but he made one series which deals with similar themes. Paranoia Agent is about a group of disconnected people who are each attacked by a young boy on rollerskates wielding a golden baseball bat. To say much more is to spoil it, but it's a grounded and darkly comic 13-part series dealing with how guilt, shame and fear trap people in roles they struggle to escape from. To give an example: its best episode is number 8, in which a group of strangers meet up to carry out an internet suicide pact, only to discover that one of their number is a little girl.
A frequent genre in anime - and one most guilty of fan service - is the 'slice-of-life' show. This is basically any animation about real people and their everyday lives, as distinct from being about spaceships or samurai or the devil, and they tend to be slow romances about dull boys pining over dull girls. Azumanga Diaoh is different in that it's focused on five female leads, each a high school girl, and their friendship with one another. I don't even think there's a single male student in it.
And while those five characters are inches away from conforming to certain standard tropes, each sidesteps that comfortably within an episode or two. Most importantly, the show is always told from their perspective, and never for the benefit of an assumed male audience. That means that even when a creepy teenaged-girl obsessed teacher appears as a tertiary character, the show finds ways to make a joke of him without it being at anyone else's expense.
It's funny, occasionally surreal and a simple, sweet show.
I watched Death Note's first episode and gave up. It's about Light, a student who finds a book on the grass outside his high school. On the inside are a set of rules: any human whose name is written inside shall die; you must be able to picture that person in your mind, so that people with the same name are not affected; if the cause of death is written within 40 seconds, that will come true, otherwise they will die of a heart attack; after writing the cause of the death, you have six minutes and 40 seconds to write further details.
Light finds the book. He's skeptical, naturally. He tries it out, discovers it works and then… Decides to use it to rid the world of evil, and spends the first episode writing the names of every criminal in prison he can find. He kills hundreds.
It was months before I went back and watched the second episode, which introduces what makes the show work: a second character, L, a mysterious detective hired to work out why all those criminals are dying suddenly. No one knows his real name, so the Death Note cannot be used on him. And so begins a brilliant cat-and-mouse story, in which Light and L try to outwit one another, and in which every twist and turn remains satisfying because it spills forward from the five rules I listed in the first paragraph. There's something almost Asimov-ian about how the logical machinations of its world and the characters actions within it. Like a very strange, deranged Jonathan Creek episode (the first season) strung across an entire series.
It loses its way towards the end, when the show overstays its welcome, but there are twenty-odd great episodes.
Serial Experiments Lain
I checked quickly and Wikipedia describes Lain as an "avant-garde anime series," and that's about right. I've watched it twice and I don't wholly understand its plot. It's about a young girl in suburban Japan who is introduced to "the Wired", a computer network similar to the internet, at the same time as one of her school friends commits suicide. It's slow, and moody, and a little abstract, but the first episode ends with a message from Lain's dead friend coming through the Wired, and from there it tips into a cryptic cyberpunk journey that abandons traditional plotting in favour of disconnected scenes, philosophy, and vignettes which teach real computer history. It's a difficult show, but I love it in part because of the mood and atmosphere it conjures.
There's a bunch more shows that I love. I'd struggle to pick between them if I was to select one more for the list above, as they're all great but all might struggle to meet the criteria I've laid out at the beginning.
xxxHolic - About Watanuki, a student with the ability to see spirits, who starts work for a witch in a wish-granting shop. It's present day urban fantasy, and it doesn't quite fit this list because Watanuki is definitely an anime character - albeit one with great voice acting in the English dub - but it's also lovely, as it pairs surprisingly dark episode-of-the-week wish-shop customers inspired by traditional Japanese folklore with the gradually developing friendships among its main cast. Sort of Buffy the Vampire Slayer-ish. I find it comforting in a way that probably makes it my second favourite anime overall. It should probably be on the list.
Cowboy Bebop - God, this is great. Sci-fi western with kung fu and a jazz-riddled soundtrack. It provides a good deal of the inspiration for things like Firefly. It is great at space noir, as well as all those other genres I just mentioned. It should probably be on the list.
Claymore - There are a lot of anime series with endless, endless fights, like yer Narutos and Bleaches and so on. Claymore is the only one of the subgenre that I've enjoyed. It's set in a medieval fantasy world in which certain women are merged with demons in order to be able to fight the 'full' demons. If they use their power too much, they become full demon. They are feared by the populace of actual humans. They are controlled by an order of freaky religious weirdoes and will turn on one another. I think there is one fight that lasts for maybe six straight episodes, but there's a lot of compelling drama and character work and an interesting world to explore at the same time. It should probably be on the list.
Eve no Jikan - A six-episode series of web-only shorts about a near-future (probably) Japan where androids exist and serve humans, set in and around the only café that'll serve the androids. If you like questions of consciousness or shows like ITV's recent Humans, then you'll like this. Tight science fiction and not a trope in sight. It should probably be on the list.
Puella Magi Madoka Magica - 'Magical girl' is a whole genre unto itself, and one of which I'm not familiar. Basically: young girl gets superpowers, uses said superpowers to fight monsters. So it goes in Madoka Magica for an episode or two before plot twists subvert the genre entirely and reveal the show to be something much darker. It should probably be on the list.
I could go on. Noein is pretty good and surprisingly informative about quantum mechanics, though it drags near the end. FLCL is joyful because of its animation but difficult to follow in terms of story or character. Ghost in the Shell: Standalone Complex does a decent job of building a procedural scifi detective show from the same core as the movie, but can't escape from the fact that its main character wears a leather jacket over a swimsuit and shin guards. Speedgrapher is about a man who can make things explode by taking photos of them, and features a scene in which he tries to take a photo of the moon. That's good. Mushi-shi is sombre and sometimes beautiful though I found it slow to the point of dullness at times. I've only seen around six episodes of Dennou Coil, but it's also great urban science fiction about virtual reality and augmented reality from the perspective of young children. Attack On Titan is about a world under siege from genuinely horrific, giant, naked, podgy men who eat humans like Peperami, though it's a shame not a single character has a real personality. I'll stop.