Something Phenomenal
 

I'm looking for something that makes your mouth go dry from being left open. I want to be speechless by the time I finish the very last seconds of the series, for good reasons of course. Something jaw dropping, amazing, and breath taking. I don't care about the genre, I just want something with a brilliant storyline and great character development.

Series that I really liked:
Clannad / Clannad AS <-- Honestly doubt anyone can top this one. If you haven't seen it, I highly suggest you check it out.
Code Geass / Code Geass R2 <-- Although there were plotholes and such, the show was simply epic.


Note: You don't have to suggest anime like the above, I'm just naming two anime that I thought were phenomenal. So don't go suggesting Death Note just because it is similar to Code Geass or other Key works just because Clannad is made by Key.

Hmm...for something jaw-dropping, I'd suggest Evangelion, if you haven't already seen it.

I'd also recommend Haruhi or Gurren Lagann. I think I'm a bit biased towards the former, though, so...yeah.

I wish I could mention Umineko, but the anime is DEEN-ified and, uh, more of an easy target for mocking than anything else. The games, on the other hand, are completely kickass.

I second his suggestions and add Eureka Seven, Simoun, FLCL, and Cowboy Bebop, for just a few.

I've seen Haruhi and curse my spoiler ridden self on Gurren Lagann.

Even with the disclaimer I feel odd about what to suggest, since my genre taste may not match yours.

One that I can say for sure is "Princess Mononoke". It really is a marvelous story.

That's a very subjective request there. For me, it would be Gakuen Utopia Manabi Straight, Sora no Manimani, and Kimi ga Nozomu Eien.

Phantom ~Requiem for the Phantom~
Nice action storyline with intense moments and lighthearted moments as well. Pretty deep and worth a watch.

White Album for some drama/romance. Or ef -tale of memories and ef-tale of melodies for even more drama/romance.

Is there such thing as a request that isn't subjective? Just take my descriptions and match it up with any anime that you feel would fall under that category. You don't even have to think about whether I'll like it or not.

"Brilliant storyline and great character development" = Seirei no Moribito. It's unequaled among anime in both categories, IMHO. I'd also recommend Ghost Hound for the same reasons.

Since you say you enjoyed Clannad, I would certainly check out Kanon 2006, if you haven't already. Both are KyoAni adaptations of Key RPGs, and I would rank Kanon as the superior series. I'd also recommend Sola, by the same writer as Kanon.

Cross Game

Maison Ikkoku

Starblazers

Fullmetal Alchemist

I think what you're saying is that you want some great drama. Am I right?

In that case I would recommend Planetes.

Well... I'm well aware that individual tastes can vary, but for me Clannad is topped by Iriya no Sora, UFO no Natsu and EF: A Tale of Memories. Although neither is a long anime - Iriya is just six episodes, while both Memories and it's sequel (EF: A Tale of Melodies) are twelve episodes each.

Since you don't care about genre I'm just going to list what I think are the best animes out there:

Honey & Clover
Cross Game
Planetes
Natsume Yuujinchou
Ookiku Furikabutte
Beck:Mongolian Chop Squad

I don't care about genre but I did state I wanted a great storyline. Natsume Yuujinchou is extremely episodic.

Kara no Kyoukai, flying to Japan just to watch the 7th movie dammit!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !

Naming Others that haven't been named on top of my head

Baccano- Such an awesome series
5 Centimeter per second - Makoto Shinkai
Tengen Toppa Gurren Lagann - It is TTGL
Mushishi - What to say bout this
Aria - Slice of Life Fan.
Monster - I read the manga, no problem.

n'thing Baccano!

Suggesting Mobile Suit Gundam 00, since you liked Code Geass. The fight scenes alone had my jaw dropping. It's one of the few shows I can watch again and again just because the mecha fights are so pretty.

I think Natsume Yuujinchou has a great storyline, and Natsume certainly goes through a lot of character development. But forgive me for taking the time out of my day to suggest animes to you that might not meet all of your criteria. I hope I haven't offended you by our different interpretations of what having a great storyline means. If you would also notice every other suggestion I gave you meets all of your requirements.

Bokurano?
Its dark, its different, and the characters themselves aren't developed as much as they are all used to develop an overarching theme. Maybe not stunning in the same Code Geass way(I loved Code Geass for that too), but this one was so unique for me, and seemed to hit all the right buttons.
Its about kids piloting a giant robot to fight the robot of the week....except not really. Includes some psychology, science, and pseudo-science, all mixed up.

I'll just toss Twelve Kingdoms in here since no one else has done so. You have to be patient at the outset, but diligence will be rewarded greatly.