Ergo Proxy
Beautifully dark science fiction anime set in the dystopian future in which nearly all of humanity is confined in a dome city. The plot, if you can wrap your head around it, is surprisingly philosophical and filled with twists. It's rather slow-paced, but well worth the effort. One of the few downsides is that the characters lack depth in comparison to other anime like Sword Art Online.
Sword Art online
SAO taught me that you can't judge an anime by its pilot. Or its first six episodes. Even though SAO is one of the more "Japanese" shows I've watched, it stands out as one of the best. The characters and their relationships are well-developed, the concepts are interesting, and it's pretty action-y. Come for Japan, stay for the emotionalism. You'll really get attached to the characters and become invested in their relationships, even though you might not remember their names (there's main character, soul patch, pig tail dragon girl, black man, dead girl, sandwich...).
Cowboy Bebop
A wild western-in-space-style science fiction series featuring plenty of guns, fighting, space, and multidimensional characters. This was among the first of the anime I watched, and most definitely one of the best. A classic.
Samurai Champloo
If you liked Cowboy Bebop, you'll like Samurai Champloo. There's not any space, but that's replaced by plenty of swords and feudal Japan.
Attack on Titan
Katawa Shoujo
Okay, it's not anime, but i'll be damned if I didn't binge the hell out of it (at least 40 hours in a week). It's an emotional rollercoaster of emotions. Most classify it as a dating sim, but it's more of a choose-your-own-story book or semi-interactive webcomic in a digitized manga form. It takes 5-10 hours to finish any one of the 5 (10? 12? A lot) story lines, most of which can bring you to tears at least thrice. The idea started on 4chan, which is surprising for something so in-depth and emotional. It follows the story of Hisao Lastname, who is diagnosed with arrhythmia and sent to a school for the physically disabled. You can work out the rest on your own. *contains sex scenes*
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