Incorrect. The vast majority of the airspace over the contiguous United States is controlled, though there is a lot of it where participation in ATC is not necessary for VFR flight. From 1,200 feet AGL up to 18,000 feet MSL you're in Class E, and from 18,000 to 60,000 you're in Class A. Above that you're in Class E again. In some places, usually over some un-towered airports, Class E will extend down to 700' AGL or down to the surface as marked on sectional charts. Class D airspace, as well as the center columns of C and B airspace, extend to the surface.

Class G airspace pretty much only exists below 1,200' AGL in most places, I think there are remote areas in the middle of the flyover states and Alaska where the Class E floor is higher because there's nothing there, but that may be changing with ADS-B and shit.

It is not mandatory to participate in air traffic control to fly in Class E airspace. Laymen tend to use "controlled airspace" to mean "off limits without permission" but that's not how that works; Restricted areas for example require clearance to enter but exist as a separate concept to the alphabet airspace system.

"Controlled airspace" means some part of the air traffic control system has coverage in that area and can provide traffic separation and sequencing for IFR flights. For VFR it's a little more complicated; in Class A airspace (high altitude en-route airspace) VFR flight is not allowed. Terminal airspace (Class B, C and D, found around airports) participation in ATC is required for all flights. ATC services in Class E airspace is optional for VFR and is on a "workload permitting" basis.