It seems that space-elevators have a competitor technology to race against:
A Space Fountain uses a continuous stream of electromagnetically accelerated metal pellets to hold things up at extreme altitudes using the same basic physical principles that a water fountain uses to suspend a plastic ball at the top of its vertical jet of water. […] Metallic pellets by the hundreds of thousands would be shot up to a “deflector” station far overhead, which would use magnetic field scoops to catch the pellets, curve them back down with an electromagnetic accelerator, then shoot them back down to the ground. The ground station would in turn magnetically catch the pellets, curve them back up with a powerful electromagnetic accelerator, and shoot them back at the station in one continuous loop. The pressure exerted on the magnetic fields of the scoop and curved EM accelerator by the continuous stream of pellets would keep the station aloft. […] The key to understanding the Space Fountain is that it uses a continuous, as in non-stop and never ending, stream of pellets to constantly exert pressure on the station it is holding up. Think back to the water fountain analogy; the way it can continuously keep the ball suspended on its jet of water is by continuously recycling the water; the water that falls back into the fountain is sucked up by the fountain’s intakes and fed back into the water jet in a never-ending loop. The same with the metallic pellet “jet” of the Space Fountain. […] Also, the pellets and the suspended station never actually make physical contact; the magnetic fields of the scoop and curved accelerator act as a kind of buffer, preventing any physical damage from the pellets screaming at the station at over 4 km/second. Yet the pellets exert pressure on the magnetic fields as they pass through them, and this force is in turn passed on to the physical structure of the station, holding it aloft.
Using this technique, it is thought the Fountain could hold up a full-sized, fully-equipped space station of 40 tons or more at almost any altitude, even space elevator heights of 30,000 km plus. However, the higher the station, the higher the required start-up and maintenance energy (see below.) A Space Fountain of about 2000 km height was mentioned to require the constant input energy of a modern city to maintain; it therefore may not be too economically viable to build Fountains much higher than this. For the purposes of building a tower capable of launching vehicles into orbit or creating super-large buildings for arcologies and such, however, 40 to 200 km or so may be more than sufficient height. […] One advantage of the Fountain is that once the system is set up, the energy needed to maintain it would be minimal compared to its start-up energy. The loss of momentum from gravity as the stream is shot up is exactly balanced by the gain in momentum due to gravity as the stream is shot down to the ground station again, but at no time does the total momentum of the system change. Entropy does dictate that some energy will eventually be lost over time, but this can easily be continually compensated for by auxiliary power stations at a small fraction of the energy needed for the system’s initial set-up. Thus, even if all power were cut to the pellet stream, it would still function normally for a while, and may take up to several hours for the suspended station 1000 km or more up to feel even a wobble.
Here‘s Wikipedia on the topic.
(I was hoping to add a mind-melting image, but there was nothing to be found.)