If you disapprove of the overly submissive way someone is acting — like the teacher's pet or a celebrity's assistant — call them by the formal adjective obsequious.
There are many words in the English language for a person or an action that is overly obedient and submissive. Obsequious people are usually not being genuine; they resort to flattery and other fawning ways to stay in the good graces of authority figures. An obsequious person can be called a bootlicker, a brownnoser or a toady. You can also say that someone gives an obsequious bow, a gesture that means, "your wish is my command."