Space elevator
If you don't already know what a space elevator is read it up here [1].
APM should make spaceflight a lot chaper and the needed strong material for the belt will definitely be producable. So if the dynamics of the cable can be handeled it could be tried out.
Note that some space access technologies could become superseeded and obsolete before they are devloped. It will certainly start with improvement of conventional rocketry and it remains to be seen which technology will pose the lowest initial hurdle to its implementation.
Disaster handling
The belt must be designed to completely burn up in the atmosphere when it comes down because it will come down ocasionally (think: one guy with a welding tourch). The belt is rather thin and lightweight (one gondola can barely transport a human) so this should be possible.
Saving a falling down gondola seems very hard. A very high speed and steep entry angle is present. Wheter a super amazing heat shield will be constructable to handle this extreme situation remains to be seen - one may dare to doubt that. The enormously high accelerations in the thinn shell of our atmosphere would crush any human though. Options would be: A standby rescue party or a a switch to an emty belt on the first singn of a belt rupture.
Shooting up wrongly due to band rupture is also a situation to make rescue plans for.
Sattelites
There are huge amounts of space debris in orbit.It's yet unclear how and when we clean up. None of the current designs tackle the problem of evading this debris. [Todo: check this assertion] To evade Sattelites it might be possible to pull adjacent belts locally together or make tricky swaps [a stab in the dark!]
Radiation
Damage
Continuous self repair of the load bearing structures is desirable.
[Todo: skim existing literature for radiation damage of nanotubes and the ensuing structural weakening]
Health
A space elevator moves slowly through the Van Allen belts. Magnetic shielding from the non mechanical technology path?
Throughput
A space elevator has a very limited mass throughput compared to e.g. the space pier concept.
To cover this up high parallelity is needed - a whole forrest.
