I am not certain if this could be answered in any definitive manner. I think a case can be made that each branch in its own right is extremely powerful. I would say that the power to determine constitutionality is a powerful element and one that rests solely in the power of the judicial branch. The fact that no other branch can determine this is what makes, in my mind, the judicial branch so powerful. With one determination, acts of Congress and actions of the President can be nullified in the assertion of not following the letter or spirit of the Constitution. I would constitute this as power, and something that only the Judicial branch possesses. Yet, I would say that a case can be made for all of the branches for being "most powerful," the essence of what the Framers' must have intended.
A matter of opinion, I suppose. I could argue the Executive Branch because the power is concentrated in the hands on one person, and they control the military. I could argue the Legislative Branch, since they have the power of the purse - the power to write the checks which would fund the military or anything else. I could also say the Supreme Court, since it has the final say on all challenged laws, there is no higher word, their appointments are lifetime and they are insulated from elections.
So take your pick! :-)
You can make an argument for any of them, but I'm going to go with the Presidency or executive branch. Here's why:
Congress has a lot of power. In fact, it has all the power to make laws. So you would think they have the most power.
But in our current system, with the power of the media and all, the President has the most power. He is the one person that just about everyone in the country recognizes. When he talks, the TV stations cover his speeches and pay attention to what he says.
This means that the President, more than anyone else, can decide what issues are going to be on the agenda. In addition, he has more of an ability than anyone else to get people to pay attention and listen to his point of view.