If you're a big fan of Chobits, you'll be happy to know there are many similar anime. In Chobits, the life of Hideki Motosuwa changes after he finds Chi, a personal computer in human form. Although none of these series have that exact storyline, they're all great recommendations for what to watch after Chobits.
To start, there's Dear S, Heaven's Lost Property, and Hal. Similar to Chobits, the anime Hal deals with A.I. and the future. Another anime like Chobits is She, The Ultimate Weapon, where a girl named Chise struggles in a battle between her humanity and the weapon she was chosen to become. Other similar anime to Chobits include Spice and Wolf and The Pet Girl of Sakurasou.
Vote the titles on this list up or down to help the best anime like Chobits rise to the top. Feel free to add more anime to the list if you notice anything good is missing.