One apparent exception is that the Supreme Court of the United States is not merely an appellate court. The United States Constitution says in Article III, Section 2 in the second paragraph says that "[i]n all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction." In these cases, the Supreme Court commonly appoints a master to take the facts in a case and then renders a legal decision on those facts. thought it need not according to Supreme Court Rule 17 and other law.
In order to understand the one ordinary exception to this rule, you have to understand the difference between courts that are created under Article III of the United States Constitution, the main ones being the Supreme Court of the United States, the United States Courts of Appeals and the United States District Courts and non-Article III tribunals. Article III courts are the courts which are the models for America's judicial independence. The judges' pay cannot be reduced during their time in office and Article III judges cannot be removed except by impeachment of the House of Representatives and conviction by two-thirds of the Senate. Non-Article III judges lack these protections- they are chosen for fixed terms and are easier to remove. Non-Article III courts are generally decision-making bodies for specialized agencies, for example, United States Citizenship and Immigration Services, or the Environment Protection Agency. Federal statutes often provide that the final administrative decisions of such bodies may receive their first Article III proceeding in a court of appeals instead of a district court. E.g. Commodity Futures Trading Commission v. Schor, 478 U.S. 833 (1986)