>>477013if you want laid-back, casual anime just ignore this list
1. Neon Genesis Evangelion - A sci-fi, mecha, psychological drama and is one of the most popular anime of all time and one of the most controversial. The End of Evangelion is still the best anime I've ever seen.
I recommend watching episodes 1-24 of the series, End of Evangelion, then episodes 25 and 26. Afterwards you can watch the Rebuilds, Evangelion 1.11, 2.22, and 3.33, if you want, but keep in mind they're not completely finished yet. Skip Death & Rebirth.
2. Revolutionary Girl Utena - A psychological fantasy that is quite surrealist. Each character is incredibly well-developed and has some of the most complex character relationships I've ever seen. The first 13 episodes are boring to most people (not to me, personally) as they are quite shoujo-like, but the episodes immediately after that are simply off the rails.
Watch Adolescence of Utena afterwards if you want. Avoid the manga unless you're a hardcore fan.
3. Puella Magi Madoka Magica - A mahou shoujo fantasy. The best thing of this anime is probably the story. The animation and art is also very good. I personally recommend the movies (the first two are the same as the series but with better art and some new scenes, and the third, rebellion, is a direct sequel to the series/2nd film).
4. Monogatari Series - A fantasy drama made by the infamous SHAFT. The series can be quite intimidating to new comers, but just follow the infographic and you'll be fine.
Ignore the note about Kizu. They're all released now. Watch them any time after finishing Bake.
Nekomonogatari Kuro is summed up in episode 6 of Second Season, so you can skip episode 6 if you choose to watch Kuro.
5. FLCL - Fantasy action show made by the same guys who did Evangelion, but its a lot more light-hearted. It's only six episodes, so its good if you want to quickly marathon a series.