Savvy science fiction fans and futurists should be familiar with space elevators. And, sure, those are a pretty cool idea, but the other day I read the Wikipedia article on this subject and came across the truly mindblowing idea of a space fountain: a floating platform that's kept aloft not by orbital velocity, but by a continuous stream of mass accelerated upward from the planetary surface. The station at the fountain's summit stays aloft by absorbing kinetic energy by deflecting the mass back down to the base station.
Imagine keeping a dinner plate aloft by firing a tennis-ball gun at it. Every time the plate starts to descend, the gun shoots another tennis ball. When the tennis ball hits the plate, the plate absorbs some of the tennis ball's energy, and it gets to stay in the air for another tiny fraction of a second. Now imagine that the dinner plate is a space station floating a couple of hundred kilometers in the air, and the gun is firing billions of tiny tennis balls at incredibly high speed every second. And instead of bouncing off the plate, the plate has electromagnets on it that bend the stream of microscopic tennis balls in a U, sending them shooting back down to the tennis ball gun at equally high speed.
OK, so the analogy breaks down. It's still a cool idea.
Incredibly enough, judging by the Wikipedia articles, it even seems that (apart from the fact that it requires incredibly massive, massive, gi-normous energy input) a space fountain appears to be easier to build than a conventional space elevator. For Earth-orbiting elevators, a conventional space elevator would require material with tensile strength exceeding diamond. Space fountains can be built from ordinary materials.
Now we just need a whole lot of energy. Nuclear fusion, maybe?
Speaking of energy, the space fountain article's "near-term applications" section is almost as fascinating as the idea itself:
A closed loop projectile system could be used for energy storage, similar to a very large flywheel, providing load leveling for terrestrial power grids. If the closed loop was long enough it could even be used for power transmission.
A very small-scale fountain tower could be used for constructing tall antenna masts rapidly, perhaps for news events and military operations. A larger and more permanent fountain tower could be ten or twenty kilometers tall, allowing one facility to provide radio and television broadcasts to enormous areas such as the steppes of Asia. Fountain towers might also prove to be an economical alternative to communication satellites for point-to-point television and FM radio communication between the various islands of some of the smaller nations in the Pacific Ocean. An elevator and observation platform could also be added as a tourist attraction.
Arched fountain structures similar to the launch loop could also have useful small-scale applicaions, notably the construction of bridges. Projectile-supported fountain bridges could be made arbitrarily long, without the need for support pillars anywhere along their span.
A couple of other links, closely or marginally related:
 Interesting fact: according to the Wikipedia space elevator article, diamond actually has a tensile strength only marginally higher than quartz; and neither is even remotely strong enough for a space elevator. Carbon nanotubes seem to be the best current hope.