OK, I'll make up a slightly random list of anime I consider good in various ways.
Gunbuster and Diebuster (aka Gunbuster 2)
SF. The original Gunbuster was the first appearance of bouncing breasts ("the Gainax bounce"), but is a fine show anyway. Diebuster is not an obvious sequel (it was made fifteen years later by a different team), but ends with a remarkable tie-in.
Texhnolyze
SF. A vision of the end of the world; dark, slow and remorseless pessimism. In the first episode no one even speaks for fourteen minutes, but there's been a rape and dismemberment. Not for the faint-hearted!
Now and Then, Here and There
Anti-war SF. Child soldiers, rape, and torture. But a compelling watch that doesn't falter.
Infinite Ryvius
SF. A sort of space version of the Lord of the Flies, but not as full-on nasty as that.
Monster
Mystery, some horror. Is sometimes cited as the best of all anime. Uncharacteristic as anime goes (it is set in post-war Europe, with only one Japanese character). Long (74 eps), slow, but continually compelling. The child Dieter is a great secondary hero.
Kino's Journey
Social comment. Separate stories in which Kino visits different societies, and her commentary on them. Parallels with real society are there to be found. Has a talking motor cycle...
Honey and Clover
College romance. A story of unrequited love among a group of friends. Happy, sad and bitter-sweet - and always moving. One of the relationships might seem a little strange to western ideas. Watch both series; the end of the first is by no means the end of the story, even if it satisfies.
Uchouten Kazoku (The Eccentric Family)
Humour, adventure, Japanese mythology. Backdrops are taken from real-life Kyoto. The main characters are Tanuki, who change between their animal form and human form - they are thought of as shape-shifters in Japanese mythology. There are also other mythical humanoid beings, and supernatural happenings. One of my favourite series of all.
Kyousougiga
Surreal. Another series based on real-life Kyoto - this time not just with weird overtones, but seriously weird. Worth a watch to see if you can make head or tail of it... Just watch the series; ignore the ONA/OVA versions - they are no more than earlier versions that the series expands and completes (I even skip the first episode of the series, which is essentially a summary of what is coming).
Pupa
Horror and gore. Twelve three-minute episodes - you can download it edited into a single 36min sequence, which I recommend. It's an allegory of child abuse and the horrifying effects on the abused. I think. I love it visually (some don't), but nothing approaches it for gore other than maybe Elfen Lied or Higurashi (When They Cry). My current (fixed) avatar is from the original manga.
Clannad
Emotional Overload. This is a real weepy, and some consider it to be simply emotionally manipulative. The only moe in my list, and a bit of the supernatural. You must go on to watch the second series after the first, as that's where the quality gets wound up to its peak. "The end will have you in tears, or your money back." This is based on a visual novel, but taking only one of the tracks as its main story. There are OVAs that take other tracks, but these are simply confusing.
But there are so many more I could add: Neon Genesis Evangelion, Mawaru Penguindrum, Last Exile, Anohana, Serial Experiments Lain, Vision of EscaFlowne, Gundam 0080 - War in the Pocket...