In a perfect scenario the space elevator is in Geosynchronous Orbit, it could be suspended a papers width above an egg, the egg would never be harmed and there would always be just enough room to slide a piece of paper between the egg and the elevator.
In reality there are a lot variables. The elevators orbit will be impacted by the moon (think tides). The land under the elevator will move (think continental drift).
How do you make or control the interaction between the elevator and the Earth?
If it was even possible (which I doubt) to anchor the elevator to the continental plate the forces would be much greater then Post-glacial rebound resulting in... well I don't know, but I think it would be bad.
Edit To Clarify: While it is possible that the end of the space elevator that touches the ground is is a thin flexible rope/wire that drifts around ground and absorbs fluctuation in distance between GSO and the surface like a kit string. In this question I am imagining that the elevator is solid inflexible tower, stretching into the sky.