No, the U.S. government has never issued a $3 bill.
From the U.S. Bureau of Engraving and Printing's web site:
"Three Dollar Notes
The Bureau of Engraving and Printing has never been authorized to
print a $3 note. However, during the early 1800's, banks operating
under Federal or State charters issued notes of that denomination.
These notes were printed by private contractors and were not
obligations of the federal government. There is a permanent exhibit
about the $3 bill on display in Philadelphia's Independence Hall."
U.S. Bureau of Engraving and Printing
As indicated, there were many state or bank-issued $3 bills at one
time. Here are some examples:
Rhode Island, $3, 1800, hosted by Federal Reserve Bank of S.F.:
New Hampshire, $3, 1780, hosted by Federal Reserve Bank of S.F.:
Mississippi, $3, 1862, hosted by Federal Reserve Bank of S.F.:
Although it comes from the period before the U.S. was officially a
country, the Continental Congress did issue $3 bills during the
" During the American Revolution, the Continental Congress issued a
three-dollar bill bearing the motto, "Exitus in Dubio Est," which
translates to "The Outcome Is in Doubt."
source: Federal Reserve Bank of San Francisco:
Here's an image of that $3, hosted by nd.edu:
From the web page:
"ever a $3 bill"
"$3 bill", site:gov
I hope this helps.