The president exerts the most influence
appointments to federal agency positions
federal independent regulatory agencies
convincing legislators to pass new laws
reducing the size of the bureaucracy

The correct answer for the question that is being presented above is this one: "appointments to federal agency positions." The president exerts the most influence over appointments to federal agency position. He does not only influence such appointments but the President also is declared as the Commander in Chief of the Army and the Navy upon stepping into office for four years per term. 

Over the appointments to federal Agency positions

Since Federal Agencies will be directly working under the president ( like collecting data, making data analysis, giving advice according to their perspective field and help the president to make a decision)