"Tower" may be a misleading word choice for what I have in mind. My approach to building vertical structures is a bit different, more on that later. There are two main issues that plague vertical architecture. Those are stability and weight. Tall towers are subjected to far more wind than houses close to the ground. Because of this, they wobble, potentially harming the structure itself and risking critical failure. The terrain itself is also a problem. Earthquakes are of course detrimental and the materials of the building aren't free from being worn over time. Mass becomes a problem when a critical point is reached and the structure collapses under its own weight. Usually the base starts breaking under the enormous pressure of all the material on top. Either way, not ideal.
However, I don't do "normal" architecture and have come up with an odd solution that fixes some problems but creates a few others: balloons. Yup that's pretty much about it. Imagine hot-air balloons (in this case filled with hydrogen) that are linked together by a series of cables. This creates a vertical cable of sorts that allows travel up and down. The tear-shaped balloons are arranged around the cable in an optimal configuration. Think golden ratio or honeycomb pattern. Unfortunately this means no windows, but they're structural weaknesses anyway (plus I have a fear of hights).
There's practically more balloon than structure but I would expect the design to allow extremely tall structures to be build. The segments are modular, so as to be easily replaced for repairs or reconfiguration. Don't want that middle segment? Link the top and bottom segment together and untether the segment in the middle. Simple.
The problem with this: wobbling. Balloons unfortunately are light and have a lot of surface area compared to volume, which makes them bad at "holding their ground" against the wind. You could mitigate this with support cables to some extend but only close to the ground. The higher we get the more "wobbly" the whole structure will become, though the mass problem has been solved (the higher you get the bigger the balloons). All this movement will put more stress on the cables and risk it breaking apart. Of course you could decide to build this inside a mountain range for protection against the wind.
[Another idea is to build clusters of "balloon-towers" so that the mechanical stress is evenly spread out (cables everywhere!!!). The towers in the middle would be more stable, while those on the outside would get more turbulence. Much like the Roman turtle formation. This may also be beneficial as piezoelectric materials can generate power using the wobbling. This design would wear down quickly but the modular nature of the structure would make it easy to fix.]
Basically, I am asking if there is a means of making this design viable.
The goal right now isn't to make a living space or anything. For now, building as high as possible is the only priority. Cost and maintenance aren't an issue, you can get creative with your propositions. The technology level is slightly futuristic.