The idea is to build very high towers by making them lighter than air, and in particular by filling it with neon. Neon would have a lifting power about a third that of helium but would be far easier to contain. There is about 18 ppm of neon and 5 ppm of helium in the atmosphere, and they could be extracted simply by freezing out all the other components. The tower could be clad with photovoltaic cells or support wind turbines to supply the energy for this extraction.
A rough calculation indicates that a tower half to one km in diameter, consisting of ring of vertical steel cables with hoops at intervals and largely filled with pure neon would have sufficient lift to carry a significant payload and keep the cables in sufficient tension. At this radius the structure should have sufficient stiffness to not bend too far as a result of wind forces. A smaller diameter structure could be allowed to bend with the wind and effectively be a buoyant cable. The structure would be flexible and tough so that a catastrophe sufficient to cause complete failure would be very unlikely and even then the collapse would be gradual.
Construction would take place at ground level. The site would be a hill of solid rock with easy access for the materials. The hoops and gas containment features being constructed one at a time and then hoisted up by letting out the cables from a ring of winches anchored in the rock. The higher parts of the tower would be lighter because the lower atmospheric density, for example it is about 10% at 18km and 1% at 33km.
The neon extraction will need a substantial plant needing a lot of power which would be supplied by solar panels and wind turbines. These would be more efficient higher up the tower above the clouds and in more continuous winds. After the tower is at its final height there would be an ongoing requirement for neon lifting gas to replace any that leaks out or to adjust the lift. Liquid oxygen and nitrogen and helium gas would be by-products.
A tower like this is probably do-able and not that expensive, but what would it be used for?
It could support wind turbines in the jet stream or provide access to the upper atmosphere for dispersing stuff to control the climate.
An astronomical observatory could be put at the top, above most of the atmosphere, although it is likely that the tower would sway around a bit, so telescopes at the top would need to be stabilised.
A shorter fatter version, say 1km diameter and 2 high, could be inhabited on the outer surface. The living units would be very spacious, with lightweight fittings and furniture. An inhabited tower might be filled with a Neon/oxygen mixture rather than pure Neon, even though the lifting capacity would be reduced, to ensure that the inhabitants don’t get suffocated by an accidental leak of the lifting gas into the apartments.
Another use would be to construct a tube down the middle and use this to shoot vehicles into space. Maybe the tube could be evacuated above the vehicle, which would then be expelled by the pressure of the atmosphere below it.
It may be that evacuating the tube could be done by condensing the gas inside, in which it would be a descendent of the Newcomen atmospheric engine
There is a promising source of renewable energy using convection in a tall chimney. A buoyant tubular tower would allow sufficient height to get a reasonable output. The energy being tapped is essentially the same that drives hurricanes and tornadoes, so these generators would tend to drain off some of this destructive energy. Maybe they could be used to protect certain areas from storms.
Other applications for this idea are very long span bridges and domes. I guess that wind forces could be a showstopper particularly for a bridge.