There was an informal JPL study that was posted to rec.arts.sf.science a very long time ago, that found that a 100km high exponentially tapering tower could be practically built by using highly pressurized balloon tanks of boron. (Practical meaning, conceivably affordable by a superpower, with a chance in hell of actually happening.)

People have also studied using dynamically supported structures. We could probably build the Lofstrom Loop and it would bring the cost of going into LEO to space elevator levels.

http://en.wikipedia.org/wiki/Launch_loop

Still, not a chance in hell of it happening in the US. China might be crazy enough to build it.