Ghost in a shell
The End of Evangelion

Series:

Death Note
Cowboy bebop
Excel Saga
DBZ
Evangelion

runner up:
Haruhi Suzumiya

Movies:

Howls Moving Castle
Grave of the Fireflies
Whisper of the heart
Spirited Away
Princess Mononoke

runner up:
Pom Poko

For movies, I like pretty much anything by Miyazaki (Princess Mononoke at the top of the list)

I haven't had the chance to see that much anime. I'm currently watching Naruto (japanese w/ eng. subs), and after that I'll watch Kenshin.

Personally, I think the best are My Neighbor Totoro, Grave of the Fireflies, Akira, Ninja Scroll, Princess Mononoke, and Spirited Away ...

Cowboy Bebop
Ghost in a shell
akira
Evangelion

Eh favorite anime. Why not... despite the fact that i know nothing but the ones brought to america.

1. Yu Yu Hakusho - lots of fun, plenty of good fights. Doesn't suffer as much from the "Only the main hero can beat anyone of importance" tendency most animes have which piss me off to no end because often times the side characters are better.

2. Ghost in the Shell TV show. (The movie just didn't hold together because they tried to mesh 3 plots into 1.)
The show has some amazing plots and is the one anime you have to give your full intention if you want to kno whats going on.  Even then all the political intrigue i'm guessing goes over everyones heads.

3. Paranoid Agent - Creepy, weird show with a very complex plot. The best episode was the one that ties in the least with the rest of the series about the three people who go around trying to kill themselves.

4. Bobobowhatever - It's surprisingly really hilarious despite the fact that 90% of it has to have been rewritten because of the cultural differences and the fact that americans wouldn't understand most of the refrences.

5. Ruroni Kenshin - Why not, it had a cute setup, some decent fights, hilarious theme song.

Other then those 5 shows... I watch a lot of anime... but more to laugh at it then enjoy it in the conventional sense.  Well actually Big O was good too... though I really don't understand anything about it.  I mean i have theories but even online nobody understands what the heck is happening.

INUYASHA!!

Thats all I have to say everything else can burn lol

Tenchi Muyo (and Universe - not Tokyo)
Princess Mononoke (and Miyazaki films in general)
Akira
Cowboy Bebop
Ghost in the Shell

I also think we need to separate this into 3 categories: Most Popular, Personal Favorite, and Best Overall. I realize that there's significant overlap, but let me explain:

On the subject of favorites: I love Dominion Tank Police to death, but I can look at it rationally and say, "That does not deserve to be on the short list of best anime ever created."

On the subject of popularity: Let me use as examples the two most popular anime of all time, DBZ and Sailor Moon.

DBZ: I did watch this show, but it had two major drawbacks besides the ridiculous power levels and inflation rate of same.
1. One could literally watch every third episode and not miss a thing. They start off by telling you in detail -- with clips -- the events of the previous episode, then do the episode, then tell you in detail -- with clips -- the events of the next episode. At least in the Toonami version.
2. The story usually took many episodes to move forward at all. I vividly remember that the two deadly androids 17 and 18, who were a planetary menace, spent at least a dozen episodes roaming the countryside in a pink truck. I think it was an ice cream truck.

Sailor Moon: Most episodes followed a formula revolving around the Monster-of-this-Episode, which often had at least one of the Senshi (in civilian form) -- or a love interest -- in dire straits, before the others saved the day. The transformation sequences were also ridiculously long, at least in the dubbed versions I saw.

Now, none of this makes them BAD anime -- on the contrary, there is a reason for their popularity, and it's not camp value. But it does, IMO, keep them off any kind of short list for "Best anime ever".

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So, to recap:

Best Overall:
Tenchi Muyo (and Universe - not Tokyo)
Princess Mononoke (and Miyazaki films in general)
Akira
Cowboy Bebop
Ghost in the Shell

Personal Favorites not included above:
Dominion Tank Police
Birdy the Mighty

Most Popular not included above:
DBZ
Sailor Moon
Ranma 1/2
Slayers
Evangelion

[edit: GitS beat out Evangelion for Best; dropped Naruto and YuYu Hakusho for Evangelion and Slayers in Most Popular.]

I'll divide it into shows and movies:

Animated Series
Eureka 7 (See avatar, I absolutely love this anime and so should you)
FMA ('Nuff said)
Ouran High School Host Club
Melancholy of Haruhi Suzumiya
Busou Renkin

Animated Movies (Okay, so it's all Miyazaki, but let's ignore that)
Princess Mononoke
Princess Nasake of the Wind Valley
Howl's Moving Castle
Spirited Away
*insert five more Studio Ghibi movies*