Random recs which you may not have heard of.
Odd Taxi - extremely fun and original new show, a psychological mystery with an engaging storyline that is simple enough to follow along but complex and thoughtful enough to make you think. When everything fits together you'll probably agree on how well-written the story actually was.
Hikaru No Go - I have never really enjoyed shows about games or sports and the competitive world behind them, except this particular one which centers around Go, a board game that I never played. Remains engaging the whole time and also is somewhat educational about this game and its role in Japan.
@sportscoach if you haven't seen it I think you'd probably like
Shinsekai Yori (From the New World) - a story which starts somewhat typically about kids with superpowers before it takes a very creative turn and focuses on the society itself, making you wonder who the villains are in it and how it should be fixed. Lots of mystery and intrigue
Terra E... (Toward the Terra) - similar to Shinsekai Yori but in more of a science fiction setting. Really just engaging plot and characters that you may love to hate, with some interesting statements on pitfalls of humanity
Thunderbolt Fantasy - From the mind of Gen Urobuchi (Madoka Magica / Psycho-Pass), not technically anime but a collaboration between animators and puppeteers, which may turn people off at first glance but offering a funny, clever, and action-filled adventure, with a similar "over the top" feel to JoJo's Bizarre Adventure
I'll try to think of any others.