The beamed power method gets about 20%, and works for humans (no g load). Escape Dynamics is trying it: http://escapedynamics.com/edispacelaunch/.

The difficulty here is more technical/scientific than engineering. Synchronizing microwave beams is hard, and if you don't you get destructive interference. I don't know if they have solved that problem yet (there have been a few proposed solutions, we'll see if any work).

Edit: Yes, rail would work great on the moon/mars. Actually there was a movie (Moon, with Sam Rockwell) where fuel is harvested on the moon and shot into orbit. Fun, crazy movie.

I think the oil derrick idea was a bit crazy (why make the problem harder than it has to be?), but the concept is sound provided that the rockets are fairly inexpensive.

I think the idea of launching to an arbitrary orbit appeals to engineers, but economically it would be more efficient to launch to a common orbit, and use cheap fuel to change orbits later if necessary. Of course, in-space refueling is only in the concept stage (but getting solved!).

To get above 6km/s, the only way I know of would be to use induction tracks (see Lawrence Livermore patent in doc). The original problem with the EM methods was getting high power switches to work repeatably. They used to cost a fortune, and last only for 10s of cycles. However this problem was solved by a completely different industry in the last couple of years (connecting wind power to the grid), and now you can get them cheap on Alibaba.

The major difficulty, actually, would be: 1) Sound pollution (not many places have high mountains, easy industrial access, and no local population). 2) Hypersonic drag (which is quadratic with velocity at that reynolds number). This is half as hard at 6km/s than 8km/s.