Chobits (2002)

In the future, personal computers have developped into "Persocons", mobile computers that look like human beings. Hideki Motosuwa, a prep school student, desperately wants to buy a Persocon but cannot afford to. One day he finds a Persocon that has been thrown away and decides to keep it. When he turns on the Persocon, all she can say is "Chii" so he decides to name her that. After a while it starts to become apparent that Chii is more than an average Persocon. Having much better performance, it seems that Chii might be a "Chobit", a type of advanced Persocon rumoured to have independent thought.