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There's weird stuff in regular movies and shows and comics too. Anime and manga have endless amounts of genres, so... whatever you're into, you can find something for you. It really depends. Imagine posting on /lit/: "Which book should I read?" But I'll try to give you some general recommendations.
Non-weebs usually cite Ghibli films as being good, though those are movies and not anime. They work because they don't rely on anime tropes, and feel/look "realistic" and use humanistic relatable themes. I've even watched a lot of their lesser-celebrated films and I think I like those ones more; the ones without fantasy elements. I keep coming back to Whisper of the Heart. There are plenty of other big-name popular anime movies, but that is it's own subject.
Paranoia Agent is a good anime, a kind-of crime mystery thing I guess. The director is famous for his movies, some people love them, but I only like Paranoia Agent. It follows a series of connected stories instead of one obvious cookie cutter plot.
The Tatami Galaxy is pretty and interesting though I think it does have sexual parts to some degree.
The anime starter kit is usually Cowboy Bebop, Evangelion, and shounen of choice. I don't like Bebop but some people love it, you should try it out. Usually non-weebs watch it and then never watch another anime ever again. I wonder if Death Note is still popular - it was a lot of fun to read once. These all still rely on anime tropes though, including sexuality.
>>38238
'Slice of Life' has become a bit too popular as a genre; so when one searches for it, the options are diluted and will contain crap. The real 'slice of life' list probably only has like five actual series.
Seconding Non Non Biyori though. That one surprised me. On the surface, seeing only the cover, I would have just passed on it. Since I grew up in the middle of nowhere, it really resonated with me.
Yokohama Kaidashi Kikou is another cool calm non-drama series. A lady runs a coffee shop in the middle of nowhere.
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