Okay, so real recommendations for real now:
The Wolf Children Ame and Yuki
A story about a mother and how she takes care of her very unusual children, from the director of The Girl Who Leapt Through Time. Features great use of editing and montage, and tells a beautiful story about being a parent.
Time of Eve
Independently produced by director Yasuhiro Yoshiura's Studio Rikka, Time of Eve is a science fiction story about a near future in which robots and humans coexist. Normally, robots are treated as property and must wear a ring above their heads at all times to signify their status, but at a certain cafe robots and humans mingle indistinguishably. Time of Eve separates itself from the norm with its unusually serious treatment of its themes, and Yoshiura is unusually talented at blending traditional animation with CGI.
It's available in both OVA and movie format. The movie has additional scenes not present in the series.
The Wings of Honneamise
Gainax's most mature work, Honneamise has a well-earned scifi classic status. Featuring one of the most intricately and thoroughly well-thought-out worlds, it tells the story of the space race in an alternate Earth filled with entirely different, though vaguely familiar cultures. But what really makes the story work is that the characters and the world they inhabit is just as complicated and flawed as ours. It examines in depth the question of whether or not it's possible for humanity to better itself when even its grandest plans are borne from selfish motives. It's also full of stunningly beautiful art.
Haibane Renmei
A thoughtful and contemplative series examining religious and spiritual themes. It tells the story of Rakka, a girl who wakes up with no memories in a town called Glie, where she meets the Haibane, angelic-like adolescents with halos and charcoal-gray wings. She discovers she herself is also a Haibane, and attempts to discover the truth of her past and the world she now lives in, a walled-in rural countryside where humans and Haibane live peacefully, but none are allowed to leave.
Accompanied by Kou Otani's masterful, moody score, Haibane Renmei grapples with a number of serious and somber themes without losing the warm touches of humanity and laughter that make it enjoyable to watch.
Bunny Drop
A touching series about a salaryman who unexpectedly becomes a single father, and the trials and joys he experiences as he discovers what it means to raise a child.
The Tatami Galaxy
This one might be a bit out there, but it does feature breakneck dialogue, surreal imagery, and heartfelt comedy. Might as well give it a shot.
Monster
A serial thriller set in '90s Germany, it tells the story of a genius Japanese brain surgeon and the boy he regrets saving. Starring Mumei's avatar.