The free atmosphere lies above what height?

The top of the atmosphere is not well defined. Rather, the gases (primarily nitrogen and oxygen) comprising the atmosphere gradually thin the higher one goes. Air pressure is 100 kPa at sea level (14.7 psi) and falls to 50 kPa (7.25 psi) about a mile above sea level. Half the weight of the atmosphere is within the first 3.5 miles (18,000 feet) of sea level. At 15,000 feet (3 miles above sea level) atmospheric pressure drops to 10 kPa (1.5 psi). 90% of the atmosphere is below 52,000 feet (10 miles). There is still enough air to support jet aircraft above 30,000 feet, and high altitude balloons will ascend to over 100,000 feet (nearly 20 miles). At 20 miles above the earth the day time sky turns quite black, and you can see the hazy boundary between air and space.

For most purposes you could say it ranges anywhere from 20 to 100 miles.