Like most people I first entered the anime fandom through the Dragonball series, followed by a long love affair with Adult Swim and Toonami anime. It hasn't been until the last year that I've had a fast enough connection speed to actually download anime and the selection at my previous library branch was pretty slim. Luckily, both of these problems have been solved and I now chug through anime by at least one series per week.
I'm partial towards dark, psychological animes with a focus on melodramatic character development. Though I couldn't exactly fit them onto my top five list Now and Then Here and There, Elfen Lied, and Boogiepop Phantom are all within my top ten. If you know anything thematically similar to these animes please feel free to recommend.
Note: By far my favorite manga stories can be found within Berserk's "Golden Age Arc" and the horror manga Monster.