Answer:
This is one of the mysteries of the universe. The answer was revealed to me once in Shanghai.
A crane "grows" or gets taller as needed as the building under construction is built. the top section of the crane as a lift built in. The crane hauls a new section, like a section of scaffolding, up to the top and sets in place.
I'm told that a crane is often anchored in place in the base of the elevator shaft, so when the building is complete, they lift the crane out section by section, leaving the very base in the elevator shaft permanently.
Or at least that's the way I understand it to happen.