From what I've read, those can be built with existing materials (which is not to underestimate the engineering challenges, which would still be great), while we still don't have bulk materials suitable for a space elevator (graphene and carbon nanotubes are strong enough, but producing them in the length required would be very hard, maybe impossible).

Alternatively, a laser or microwave launch system that works by transferring energy from ground-based lasers or microwave arrays to the spacecraft (it could use air for reaction mass at lower altitudes, switching to on-board propellant as it get higher up). The advantage there is that you don't have to lift the power plant itself.

For non-human payloads that can handle extremely high accelerations, some kind of railgun might do the trick.