The U.S. is a single-currency nation, and the Fed has the sole authority to print the national currency, the U.S. dollar. It wasn't always that way, though. The National Banking Act established an effective national currency, but it would still be another 50 years until the birth of the Federal Reserve System (The Fed lists a snapshot of its history and the US dollar) to monitor the health of the banking system and regulate the money supply.
Anachronistic: I think of the mid 1800's when chartered banks were issuing their own notes when I see this video report of a region in Western Mass. that issues its own currency for trade, the "Berkshare".