Hamilton's National Bank was established.

There was a National Bank from 1791-1811, then no bank for 5 years, then a new National Bank from 1816-1836, then no bank for a while, then a new National Bank from 1863 (Civil War debts and loans and what-not) until the early 20th century when the Federal Reserve banks were created.

Here are the Federal Reserve Bank of St. Louis's archives about the original National Banks (1791-1811, 1816-1836). http://fraser.stlouisfed.org/banks/

You should try /r/HistoricalWhatIf for this. You can get answers here, but this is more appropriate over there.

This question is more suitable for /r/HistoricalWhatIf.