When is each branch (legislative, executive, judicial) most powerful? what situation would they be most powerful in?

It seems as if the the folks who have answered think that it is clearly the Judicial Branch, but they couldn't be more wrong. If anything, the Judicial Branch has the least amount of authority in the federal government. The Supreme Court, in fact, has only implied power. The High Court acquired the ability to deem a law unconstitutional via one of their own rulings in 1803 (Marbury vs, Madison); nowhere in the Constitution of the United States does it authorize the power of judicial review to the Supreme Court. Furthermore, the Court can only rule on cases brought before them by parties with a standing concern (i.e., individuals who are being directly and immediately affected by the law). If the case never reaches their courtroom, all the Supreme Court can do is sit idley by and watch from the sidelines. As far as the preconception that a Court ruling not being able to be overturned, that's not entirely accurate. The Congress, from time to time, can regulate and make exceptions to Supreme Court decisions, in particular where it pertains to the Court's appellate jurisdiction. 

You would have to list the the Executive Branch as a pretender as well, considering that The Congress can supercede nearly every decision that the President can make. His veto powered is neutered by the fact that Congress can simply override and institute any law that the President strikes down, Presidential appointments are subject to approval by both Houses, and his job security is determined largley at their discretion. 

The Legislative Branch, without a doubt, holds all the cards in this discussion.They write our laws, levy our taxes, declare our wars, confirm our Supreme Court Justices and Cabinet Officials, decide how the federal money is spent, amongst a plothera of other actionable powers. In addition to their great many capabilities, as it concerns this discussion, Congress has the authority to impeach and dismiss the occupants of the other 2 branches. This, above all else, makes the Legislative Branch, undoubtedly, the most powerful of the three.