Proposals were made around 1000 km altitude at the lower tip. Boeing proposed a 600km tether with a transfer altitude of 100km (at mach 10), and later increased that to 150 km. The Lockheed U-2 is one of the highest flying airplanes ever, our main cold war spy plane. It flew twice as high as a passenger jet, and cruised at 70,000 feet, just over 21 km.

Planes would not get tangled in it.

Thanks. The GP's comment "attach the string at, maybe, the equator of earth" is what confused me, thinking the string is attached to the earth, with the weight in orbit.