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Neon Genesis Evangelion
One of the most important anime of all time, and still very relevant.
The Melancholy of Haruhi Suzumiya
It was super relevant, and is still a pretty important anime to the anime community that should be watched, constantly referenced. Has some interesting storytelling gimmicks, and ends with a movie that a lot of /a/ considers the greatest masterpiece of all time, and overall just interesting.
Watch it in broadcast order, not chronological. After you finish the first season, you can continue onto season 2, or if you want to put a lot of effort into it, you can wait for a few months and watch everything in chronological. But broadcast order is the way to go for first time watching season 1. Some people will tell you to skip a bunch of episodes, but they are all important for building up a masterpiece, and helps understanding one of the greatest events in anime community history. Don't skip any episodes. Finish with the movie.
Serial Experiments Lain
A classic masterpiece, and probably the most mainstream of the deep, confusing anime. Very good anime, watch it. Apparently it used to be an important part of 4chan culture too.
Azumanga Daioh
Another classic, dominated the internet back when it was newer, still a great comedy to watch.
