 If you liked
Chobits
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 ...then you might like
Elfen Lied
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A girl is found -> she is taken home -> she is innocent, childlike, and she cannot speak but one syllable (so she is named) -> she is found to be unique of her kind, and seems to have a darker "version of herself" lurking somewhere deep in her subconscious.
(Note though, that Chobits does not delve into the level of gore Elfen Lied does, and Elfen Lied, though it does contain nudity, does not focus primarily on ecchi as comic relief like Chobits.)
In both Chobits and Elfen Lied, a boy finds a girl randomly and takes her home. The girl can't say anything other then one word so she is named after that one word (Chi in Chobits, Nyu in Elfen Lied). They boy then starts to find out more about the girl and uncovers some dark secerets. Also they both have romance.
The first impression I had of Chobits when I read through to around volume 2 or 3 was that it was really similar to Elfen Lied. Both are high on sci-fi and strange happenings and both involve a robotic girl who lost her memory meeting a boy at the start. And in both cases, the girl has sort of a dual personality. But the story of the 2 are quite different afterwards.
I also recalled Kanon(2006) anime, which uses a flashback method to show the storyline.