It's hard to believe but there was a time in the United States (1837 - 1863, also known as the Free Banking Era or Wildcat Banking) when there was no central bank and paper money were issued by a variety of private banks.
Unenumerated Blog has a neat collection of bank notes from this freewheeling era of banking:
During this era the U.S. had no central bank and paper money was issued by a variety of private banks. Some was even issued by manufacturing and retail companies. This money was backed by gold, silver, real estate, stocks, bonds, and a wide variety of other assets. You can no longer cash them in, but they are now worth often substantial sums as collectibles. As you can see below, the note designs were more varied and creative than modern money, and were remarkably free of politicians' faces.