Which is the most powerful branch of government? Why? Give examples.

In my opinion, the most powerful branch of United States government is the legislative branch, Congress, in spite of the fact that it has not accomplished all that much in recent years.  I think it was intended to be the most powerful branch, and that this is reflected in its being created as the first article of the Constitution. We were intended to be a nation of law.

Congress has an extensive list in Article I of areas in which it can create law.  Within those areas, immigration, interstate commerce, taxation, and war, to name but a few, it can create laws that can withstand attacks by the other branches of government.  If a president vetoes a law, Congress can override that veto. If the Supreme Court finds legislation wanting somehow, Congress can tweak it to make it pass constitutional muster.  For example, the Supreme Court, last term, I believe, struck down a section of the Voting Rights Act, and Congress, had it the political will to do so, could have amended the act to restore that section while still being constitutionally compliant.  It has declined to exercise its power this way for political reasons, but that does not mean its power does not exist. 

Congress also has the responsibility to advise and consent on federal presidential appointments.  We have seen during the past two presidential administrations its ability to hobble the executive branch by simply failing to move along on that process, most strikingly, right now, as it declines to consider a proffered candidate for the Supreme Court. That it is constitutional for Congress to decline to consider a presidential nominee has never been subject to challenge in the courts before, but at the moment, it is exercising this power, which, at the very least, seems to be implicit within the Constitution. 

No matter which other branch of government wishes to oppose the will of the legislative branch, it does have the means of overpowering that opposition. Whether it chooses to exercise that power or not is another matter. It is also worth noting that in a nation in which the majority of eligible voters choose to not vote, the most powerful branch of government by no means represents the majority of people.