How large is the Underground Economy in the US, and is it growing or getting smaller as a percent of GDP?
There has been some research on this. For example, see the papers by Cagan (1958), Tanzi (1983) and Schneider (2007), Schneider et. al (2010).
If you would like a survey kind of thing, there is a staff paper from St. Louis Fed: Restrepo-Echavarria (2015). Link: https://www.stlouisfed.org/publications/regional-economist/january-2015/underground-economy and also Schneider and Enste (2000).
Based on Table 3.3.3 (page 24) of Schneider et. al (2010) and the discussion (page 17-19 of same paper) which makes me believe that the numbers on the table are percentage of GDP, USA moved from 8.8% in 1999 to 8.4% in 2007 with a country average of 8.6% in this time period.
Note that (as these papers would mention) there is significant debate about defining and estimating "shadow economy".
Cagan, Phillip. "The Demand for Currency Relative to the Total Money Supply." Journal of Political Economy, August 1958, Vol. 66, No. 4, pp. 302-28.
Restrepo-Echavarria. "Measuring Underground Economy Can Be Done, but It Is Difficult" Federal Reserve Bank of St. Louis The Regional Economist, First Quarter 2015, Vol. 23, No. 1
Schneider, Friedrich. "Shadow Economies and Corruption All Over the World: New Estimates for 145 Countries." Economics: The Open-Access, Open-Assessment E-Journal, July 2007, Vol. 1, No. 2007-9, pp. 1-66.
Schneider, Friedrich & Buehn, Andreas & Montenegro, Claudio E. "Shadow economies all over the world : new estimates for 162 countries from 1999 to 2007," , 2010, Policy Research Working Paper Series 5356, The World Bank.
Schneider, F. and Enste, D. Shadow Economies: Size, Causes, and Consequences. The Journal of Economic Literature, 2000 , 38(1): 77–114.
Tanzi, Vito. "The Underground Economy in the United States: Estimates and Implications." Staff Papers—International Monetary Fund, 1983, Vol. 30, No. 2, pp. 283-305.
Massive, and only going getting bigger with the popularity of cryptocurrencies in black markets. As overpriced and speculative of an asset as they are, it's allowing shadow value flows like never before in modern tax systems, and even more so in the world's most impoverished and destabilized economic systems such as Venezuela (not the Petro). There are very big players in Bitcoin, Monero, etc, and in some form, cryptocurrencies are going to stay around for a while.
It's pointless trying to estimate the exact share of hidden transactions in an economy, but it has certainly grown since the post-WWII economic and information boom.