I would have to say the Executive Branch is the most powerful today, even though the system is designed for the three branches to be roughly equal.  The power of the President really expanded during George W. Bush's tenure, and the ways in which both Executive Orders and signing statements have been used recently represents that expansion.  So does the Patriot Act and increased power for executive law enforcement agencies.

While they had something like congressional approval, and continued funding, Vietnam, the Persian Gulf War and the current Iraq and Afghan Wars are all undeclared by Congress.  The President simply ordered in troops and Congress supported it, and the Supreme Court never ruled the action unconstitutional.

The founding fathers designed the government to have what is known as a "balance" of power between all three branches. So no one branch is supposed to be more powerful than the other. Is this true in reality? You would get a variety of opinions on this one. Some feel that the Judicial Branch is too powerful today because although the courts are supposed to interpret the laws, some accuse them of actually making laws in their decisions, etc. Other people would argue that the Executive Branch is too powerful today. The Congress is supposed to declare war, however, but it is the president's job to protect the people, so if he/she feels that protection entails a "police action" or "conflict", he/she can declare this without an actual declaration of war. The Congress, however, controls the money, so some people think that this branch is the most powerful. No matter what the president winds up doing, he/she must get the money to pay for it from the Congress.

There are some really good references regarding government here on enotes.

What is the most powerful branch of government?

The United States Constitution is designed in such a way that the three branches of government hold nearly equal powers. While it may be argued that one branch is more powerful than the other two, each branch may offer checks and balances on the other branches. The Constitution enumerates these checks and balances as well as the limits of the branches and the members therein. Moreover, the branches cannot intersect, and therefore, they cannot directly influence each other by sharing members.

No Senator or Representative shall, during the Time for which he was elected, be appointed to any civil Office under the Authority of the United States, which shall have been created, or the Emoluments whereof shall have been increased during such time; and no Person holding any Office under the United States, shall be a Member of either House during his Continuance in Office. (section 6, clause 2)

An example of a Congressional (Legislative Branch) check is the power to override a presidential veto with a two-thirds majority vote in both the House and the Senate. An example of a Supreme Court (Judicial Branch) check is the power to declare a law unconstitutional. Finally, an example of a Presidential (Executive Branch) check is the power to declare a national state of emergency and override the need for congressional approval for a major act.

What is the most powerful branch of government?

No one branch of government is more powerful than another, at least in theory. The Constitution set up a system of checks and balances to keep any branch from having too much power. Congress, the legislative branch, can pass laws, but the President can veto the law. Congress could override the veto with a 3/4's vote. The Supreme Court, the judicial branch, could declare a law unconstitutional or a veto unconstitutional. The President, the chief of the executive branch, can promote programs, but it takes Congress to enact a law. The President can appoint people to various positions in government, but Congress must approve the appointments. The President is the head of the armed services, but Congress passes a budget to fund these programs. Only the president can declare war with the backing of Congress. This is only a short synopsis of the checks and balances. There are many more examples.