I am going to answer what I think is the true question:
“What are the effects of the US government mis-representing the amount of currency it prints?” (The actual wording uses imprecise terms that people mis-use.)
The first thing to note is that the US Treasury Bureau of Engraving and Printing is in charge of printing banknotes in the United States (the Mint produces coins. Link to website.
I am no legal expert, but I think the following summary is close enough to the truth. The Treasury (and printing bureau) reports to the President, while the Fed is an independent body created by Congress. All are expected to follow laws set by Congress. The US Constitution says that Congress regulates coinage. As such, any monkeying around and not following legal procedures would be a really big issue.
But let’s pretend that they want to get around rules, and somehow get the hundreds of people who would see what is happening to remain quiet. What would happen if they print more banknotes than they say they are?
You immediately run into the problem that the Federal government - nor any legitimate business - pays for things using large amounts of banknotes. Any attempt to do so within the United States would just end up being returned to banks, and so the money would be “un-printed” as the banks ship it straight back to the Fed.
Dumping it overseas (as suggested) will not work, since foreigners already have too many US banknotes floating around. “Money laundering” - the conversion of banknotes to more useful bank deposts - is already a weak link for organised crime. New inflows makes the problems more difficult.
Furthermore, the obvious question remains: what is the U.S. government going to buy with suitcases full of \$100 bills in foreign countries?