Here you go, the top ten anime as stated by me. Sorry, it’s a little Miyazaki heavy. 
I want to point out, though, that this is a best anime list. I did not rank it by my favorites.
1 – Vampire Hunter D (1985)
This one was one of the first anime brought to America, and while the thing isn’t perfect, it’s still a good work. The dub is very accurate for the time.
2 – Fushigi Yugi (1995)
This one is such an epic, emotional tale. There is a great span of characters so everyone has someone to like. It’s also such a catching story that I’ll bet even guys would get choked up. 
3 – Princess Mononoke (1997)
There’s a reason we watched this in my Japanese culture class. Shinto, the moral conflict about modernization, the disappearing of nature, the Emishi, it’s such a great cultural piece. The beautiful landscapes and story add to the experience.
4 – Spirited Away (2001)
I add this one because it inspired the Academy to wake up and acknowledge anime. It is also a great moral story.
5 – My Neighbor Totoro (1988)
Such a great kid’s movie. 
It’s full of great characters and is a good cross-section of Japanese country life.
6 – Ghost in the Shell (1995)
Female butt-kicking lead, techno-savvy plot and awesome cerebral elements make this one a given.
7 – The Castle of Cagliostro (1979)
Aww, Lupin is the nice guy! Hey, it works for the story. I think this one is a good blend of comedy and drama. (I also think it’s funny that it shares the principal dub cast with Fushigi Yugi. )
8 – Akira (1988)
Another one of those anime that got the US hooked, Akira is a good thrill ride. The animation is great, especially when Tetsuo is having, shall I say, body problems.
9 – Perfect Blue (1997)
You forget that this is an anime when you watch it. The plot is so nuts that it will take you a couple times to get what happened. Why they haven’t made this into a movie yet is beyond me.
10 – Death Note (2006)
This one is a good mental story. I liked seeing how Light would get himself out of the next hole he dug, and parts of the story were very unpredictable.
Woo. Got my fingers worked out on that one.