Green’s Dictionary of Slang

(US) superior, important; arrogant, pretentious.

1931		D. Runyon ‘Dream Street Rose’ in Runyon on Broadway (1954) 55: Your ever-loving wife is getting to be very high mucky-mucky in society.
1957		G. Metalious Peyton Place (1959) 160: Not even a high mucky muck doctor like you.