@chris
Well Trump isn't the boss of anyone other than the executive branch. So he could fire whole departments in theory but only the ones directly under him. So he would only be weakening his own executive branch by doing so, but sure he could.
@adam
Doc Freemo 
@freemo
It's surprisingly extensive:
> The executive branch includes 15 Cabinet departments (e.g., State, Defense, Justice), independent agencies (e.g., CIA, EPA, NASA), regulatory commissions (e.g., FCC, SEC), and government corporations (e.g., USPS, Amtrak). Some agencies operate with more independence from direct presidential control.
@adam