What is it called when someone is addicted to plastic surgery?

There is not a term for the addiction, but there is a recognized psychological disorder that affects some surgery addicts. It is called "body dysmorphic disorder" (BDD), and sufferers have a distorted image of their own appearance. This is sometimes manifested as disapproval after surgery is performed, leading to another surgery to correct the apparent flaws. Because of the high costs of plastic surgery, this disorder is usually apparent only in the very wealthy.