As Kennita and others have pointed out, it is hard to build such a
structure. However, even if you could build it, it wouldn't be that useful
for getting to space. It is a non-obvious fact that most of the difficulty
in getting from here to orbit is in acquiring speed, not altitude.
Attaining orbital velocity of 8 km/sec requires storing 16,000,000 joules
in every kilogram of payload. On the other hand, reaching an altitude of
100 km requires only 1,000,000 joules per kilogram. So climbing to the top
of a 100 km tower gets you only 6% of the way to orbit.
--CarlF