Apart from Ghibli Movies, also Satoshi Kon Movies, like Perfect Blue and Paprika.
Original Ghost in the Shell
The Girl Who Leapt Through Time
In This Corner of the World
Kotonoha no Niwa
...Most movies that aren't part of a series should be fine, really.
As for anime series...
Serial Experiments Lain
Kino no Tabi (I've only seen the 2003 version, so idk whether the new one is the same)
Mononoke (not the ghibli movie)
Hunter x Hunter
NHK ni Youkoso
Darker than Black
Kiseijuu: Sei no Kakuritsu
Fullmetal Alchemist Brotherhood
Michiko to Hatchin
Sayonara Zetsubou Sensei
Mecha, but maybe also The Vision of Escaflowne
Devilman OVA (The birth 1987 and demon bird 1990) (manga is better)
Devilman Crybaby (is kinda different from the original manga)
Tiger & Bunny
...I'm really not as sure about those last ones, but they might fit the bill as well.
Also I'd recommend checking out the ones I put higher on that list first.
Hunter x Hunter might seem like a typical shounen at first, but imho it gets better the more you watch.
Also probably these, althogh I haven't actually watched them yet:
Shigatsu wa Kimi no Uso
Higashi no Eden
Legend of the Galactic Heroes
Shoujo Kakumei Utena
Shouwa Genroku Rakugo Shinjuu
Eizouken ni wa Te wo Dasu na
The Tatami Galaxy
Ping Pong the Animation (It's sports but I heard its really good)
Bakuman (Only watched a few episodes and I really hate this one, but I guess there are people who like this)
If you ever feel like watching something truly horrible then Chargeman Ken, other anime will look good in comparison.
There's more in picrel, just watch whatever catches your interest.