What is the compensation for an entrepreneur in residence?

You need to be a little more specific... what type of entrepreneur (entity), what line of business, is there a profit, what are the owner's goals, etc. A sole proprietor takes no compensation (typically). For an owner of a Corporation or an LLC, it would depend on the circumstances, there are various tax and personal reason you would do certain things.

By "in residence" are you referring to a Doctor performing a residency, or do you mean by an individual operating a business from their home?