Final answer:
The 'hat' that does not represent a role played by the President of the United States is 'Chief Justice', as this role is held by the head of the Supreme Court.
Step-by-step explanation:
The role among 'Chief Executive,' 'Commander-in-Chief,' 'Chief Justice,' and 'Chief Legislator' that does not represent a 'hat' worn by the President of the United States is C) Chief Justice. The President serves as the head of the executive branch, as Chief Executive, and holds the role of Commander-in-Chief of the Armed Forces. The position of Chief Justice, however, belongs to the head of the Supreme Court and is not a role performed by the President. As Chief Legislator, the President has the ability to influence law-making processes, but cannot legislate directly. The President also exercises significant influence in foreign and domestic policies.