Other than the Supreme Court and Congress, Is there any position in the US Government that is not required to answer to the President of the United States?
You've actually tripped into an interesting constitutional debate, albeit one that was settled pretty definitively by the Supreme Court during the Reagan years. In Morrison v. Olson, the Supreme Court ruled 7-1 that Congress can create executive offices with executive powers that cannot be fired or answer to the President.
The controversy behind that case was the creation of an Independent Counsel named Alexia Morrison to investigate Ted Olsen, Assistant Attorney General. Olsen argued that the office was unconstitutional, as it gave executive power to individuals who did not answer to the President even though the Constitution grants "[t]he executive Power" solely to the President. Earlier in Myers v. United States the Supreme Court had ruled that the power to remove executive officers, even those who had to be approved by the Senate, was an essential part of the President's executive authority, and cannot be infringed upon by Cogress.
The ruling by the Supreme Court was that because the Independent Counsel didn't report to Congress or the judiciary, its existence was not a violation of the Constitution's separation of powers:
The Act, taken as a whole, does not violate the principle of separation of powers by unduly interfering with the Executive Branch's role. This case does not involve an attempt by Congress to increase its own powers at the expense of the Executive Branch... Similarly, the Act does not work any judicial usurpation of properly executive functions. Nor does the Act impermissibly undermine the powers of the Executive Branch, or disrupt the proper balance between the coordinate branches by preventing the Executive Branch from accomplishing its constitutionally assigned functions. Even though counsel is to some degree "independent" and free from Executive Branch supervision to a greater extent than other federal prosecutors, the Act gives the Executive Branch sufficient control over the independent counsel to ensure that the President is able to perform his constitutionally assigned duties.
Justice Scalia, the sole dissenter, is still adamantly opposed to the Court's ruling in Morrison. Before his appointment to the Court, Samuel Alito was very openly critical of the decision for its "elastic language" that "can accommodate, unfortunately, quite onerous restrictions on the president's power." I've never heard of Roberts, Thomas, or Kennedy indicating how they felt about the law, but I wouldn't be surprised if at least one of them shared their skepticism in light of the traditional conservative support for a strong executive branch.
So yes, there are offices of the US Government that do not answer to the president. But there is significant opposition to the arrangement among some legal elites.
Depends how you define "Government".
Federal Reserve board and its chairman qualify. They are appointed by the President but aren't controlled by or firable by him subsequently.
Once appointed, Governors may not be removed from office for their policy views. The lengthy terms and staggered appointments are intended to contribute to the insulation of the Board--and the Federal Reserve System as a whole--from day-to-day political pressures to which it might otherwise be subject.
So does Vice President of the United States since he's elected and can't be removed by the President.
An interesting possibility is Independent Council, but I'm unsure if they are Executive branch in the first place. Will dig but probably not.