Why do two adverbs follow come in the phrase, come on in? I know come in, come on, go away, but when I hear "Come on in" in American movies, I can't figure out its grammatical structure.
Come on in is the longer version of come in. It's an idiom as mentioned by WS2, which means proper meaning cannot be deduced from individual words. Here's the full description of this particular idiom:
Come (on) in. and come on in(to) something Enter.; Come into this place. (A polite invitation to enter someone's home, office, room, etc. It is more emphatic with on.) Bob: Hello, you guys. Come on in. We're just about to start dinner. Bill: Come in. Nice to see you. Mary: I hope we're not too early. Bill: Not at all. Come on into the house and have a cold drink.