Total Recommendations: 5
Small, personal stories of drama and exploration set in a fascinating, "big" setting. Both are likely to leave you wanting to find out far more about the world they take place in than is revealed while following the footsteps of only a few people in it and their limited viewpoint. Dennou Coil is soft sci-fi; Haibane Renmei is spiritual fantasy.
Fuzzy animals behaving semi-anthropomorphically in a very laid-back, slice of life with comedic moments. Chi's Sweet Home is about the antics of an energetic kitten that very much acts like a normal kitten and relies on that for humor; Shirokuma Cafe is about a polar bear running a cafe with animals that are obviously rather more human-like and relies more on wordplay. Both very family friendly, but perhaps too slow-moving for most kids.
Dystopic worlds populated by powerful people in shady, but well-intending organizations with mysterious overarching plots throughout several discrete subplots and similar levels of character attrition along the way. In both series, questionable actions from morally ambiguous characters lead to escalation or resolution, and in neither do the shows assign a simple good or bad metric. Shinsekai Yori is more contemplative and has a feeling of slow dread; Book of Bantorra is packed to the brim with action and takes you along for the ride.
This may seem like an odd recommendation at first glance, but it's not completely without reason - both invite contemplation on moral issues, the characters, and the mystical world they're set in without making any judgment, welcoming the viewer to either make their own or simply accede that there is no clear right or wrong. Shinsekai Yori is much more dramatic and epic, presenting a world that is dystopic; Mushishi is subdued and calming, presenting a world that is filled with muted wonder, if at times melancholic.
Both series deal with the supernatural as found in a cloistered community, and the drastic measures which must be taken to deal with the danger to humanity. Shiki is more cerebral and philosophical; Another is more of an excellent demonstration of classic horror tropes.