I have a few anime I love, but my top two are Phi Brain: Kami no Puzzle and Natsume Yuujinchou. The first is a bit more action-packed most of the time and is about geniuses solving puzzles that are deadly and whatnot. It has some generalized history in it (not really, but like, all of them have famous genius's nicknames, etc)...and a lot of fantastical stuff despite being in a fairly modern setting.
Natsume Yuujinchou is about a guy who can see youkai and has a book that they all want which he inherited from his grandmother, the last person in his family to have that ability. It shows him dealing with youkai, occasionally becoming friends with some and enemies with others, and even more unique characters. It's deep and a bit more calm to me, despite there being a lot of turmoil in it.
Another one I really enjoy recently is Classicaloid. It gives the classic musicians from Bach to Beethoven an android-like twist and is super colorful. They updated the music so that it's more rockish, but teaches you about the musicians in it and about the songs they updated/play. It's completely nuts and such, but it's also really vibrant and fun. And weird, did I mention weird?