Learn English – What’s the origin and reasoning behind the phrase, “I’ve got a monkey on the back”

It’s true that this expression is in an interrogative form that is not normally found in contemporary English. It uses inversion rather than the auxiliary verb do. We would normally expect What do you have? (or, in the UK at least, What have you got?) It’s a colloquial way of saying, in the Oxford English Dictionary’s definition, ‘anything else (similar) that there may be, or that one can think of’, or even, simply, etcetera. It’s of US origin, and the Oxford English Dictionary’s earliest citation is from the 'New Yorker' magazine in 1925.

I think the two meanings are so close it's hardly practical to try and split usages into one or the other. Lots of things can be called "monkey business" - in if I catch him up to any monkey business with my sister, I'll bash him, it simply means making sexual advances.

Other words with the same broad spread of meanings include shenanigans, mischief, skullduggery, roguery, nonsense, funny business/stuff, antics, horseplay. It's a matter of context whether the activity thus described is immoral, or simply frivolous, but it's invariably somewhat disapproved of.