Chobits (Vol. 1)

Hideki is your typical 20-year-old slacker. He didn't get into college straight out of high school, so he's working a dead-end job, can't find a girlfriend, and is perpetually broke. If only he had a persocon - a humanoid robot - to keep him company, perhaps things wouldn't be so bad. The problem with persocons is they are incredibly expensive, and it would take Hideki a lifetime to afford one. Then, one day Hideki finds a particularly cute female persocon dumped in a pile of trash, so he decides to take her home. As they say, one man's trash is another man's treasure...