Some of the most un-cliche anime I know of that haven't been mentioned yet. Some dark, some not.
Instead of Evangelion, watch Dual!: Parallel Trouble Adventure. Boy has visions of giant robots fighting in a parallel dimension, crosses over and becomes the pilot of the best giant robot to ever wear high heels and becomes a central character in the war that's going on. It takes all the elements that were annoying about Evangelion and makes them funny and it's a breezy 14 episodes.
The Legend of Black Heaven: Former rocker Oji has to keep it a secret from his wife and son that he's putting his band, Black Heaven, back together because their unique sound can power the ultimate weapon of a group of humanoid aliens fighting against other evil aliens.
El-Hazard: The Magnificent World, a small group of students and their PE teacher go to another world, gain conditional superpowers, impersonate a missing princess and fight against giant bugs and girl who is a living ultimate weapon. 7 episodes.
The Irresponsible Captain Tylor, Tylor seems to be an idiot, has God-like luck, joins the space force, gets put in command of a crew of misfits and makes enemies on both sides of a looming intergalactic war between humans and the Raalgon empire. One of the great anime questions: "Is Tylor an idiot, a genius, or just supremely lucky.?"
Wolf's Rain: Wolves are creatures of legend, thought to be extinct, learned to hide themselves by appearing human. Four wolves go on a bloody quest for Paradise with a girl who is a human flower. Gets pretty darn dark near the end.
Vandread, men and women live on different planets and war with each other, three men wind up on an all-female pirate ship and face off against an alien menace harvesting humans for body parts with some of the biggest mechs from the pre-Gurren Lagann anime era. For most of the series the men and women have no idea how to interact and are more often terrified of each other, avoiding most cliche harem pitfalls.
Nerima Daikon Bros., two guys and a girl work a raddish field and dream of singing to crazed fans in their own giant dome. A musical from Nabeshin, watch the dub, it's one of the greatest.