Show me someone building an airplane. Oh sorry, you need an FAA license for that...
Check out the Experimental Aircraft Association. Visit the Oshkosh Fly-In. FAA regulations on experimental aircraft are quite lenient. You can't carry passengers or fly over heavily populated areas, which is reasonable enough. For flight test, there's the Mojave Air and Space Port. "My job is to give people permission. Every day in the skies over Mojave and on the ground at Mojave Air & Space Port, people take enormous risks, which someday will yield great things for all humanity."
-- Stuart Witt, CEO, Mojave Air & Space Port.
"You want to test a rocket engine? This is a place where you can do that." -- Board of Directors, Mojave Air and Space Port. SpaceShip One and various X-Prize trials have launched from Mojave. Rotary Rocket flew from there, although not very far. I know people at TechShop building upper stage engines for orbital insertion.
There are several ultralight helicopter kits. Quadrotors seem to get bigger each year. Thrust-type VTOLs need a lot of power, which usually means jet engines, which means a flying car will cost about as much as a small bizjet, which limits the market. Paul Moller built a flying car; it doesn't work, but that's Moller's problem, for which he's been making excuses for 40 years. I had some hopes for Urban Aeronautics out of Israel, which was showing a non-flying mockup in 2010, but they never made it fly.
Government is not preventing you from doing any of these things.