The Maximum Height a Skyscraper can be Built

As to the maximum height of skyscraper, this article gives a good answer and explanations.
In the article the author proved that a skyscraper could be built to at least as tall as the tallest natural mountain, as long as the base of the structure is large enough like a mountain, in combination with a tapering shape in order to reduce the intensity of pressure on the soil ground. And skyscrapers are far lighter than solid mountains, a skyscraper is mainly composed of air, weighing only around 15 percent of a mountain, so it could be 6.67 times taller when the weight is same as a solid mountain, by this way a skyscraper could be built to nearly 59 thousand meters tall (6.67 times of the height of Mount Everest), that's 150 times of the height of the Empire State Building, how crazy it is.

However the skyscrapers are not mountains, there are many factors preventing them from having a base as large as that of a mountain.
First, skyscrapers are supposed to accommodate people, which means the natural light is a vital factor to be considered by architects or developers. If a building's base is wide as a mountain, there'll be large amount of floor space lacking of natural lights, nobody wants to live in dark rooms, maybe only suitable for some shopping malls, conference center, in which light illumination are largely used.
Second, this kind of skyscrapers will be visually unappealing, many people and social groups will protest against it.

And the most vital factor is the financial difficulties, which is also the lots of skyscraper projects are facing. This factor comes from two aspects, respectively are the scarcity of the land and high cost on the construction. On average a standard skyscraper occupies an area of 2000 square meters, in some cities like New York such a plot requires up to 500 million dollars, and the cost for construction ranges from some 300 million to over a billion, a skyscraper with a base as large as Mount Fuji equals to thousands of standard skyscrapers, which real estate developer in the world has so much money to build it? Today developers are struggling to secure the funds to build those standard size skyscrapers, and in New York City, most of newly built towers are super skinny as they are easier to build and highly profitable.

So taking the mountain as the reference to estimate the maximum height of skyscraper seems very unreliable.
We can't take those normal mountains as the reference, then how could we estimate the limit to the heights of skyscrapers?

Thanks to our amazing nature, there are some bizarre mountains in the world. Some of them appear like normal skyscrapers.
A relatively notable example is the Mount Hallelujah in Hunan, China, see the picture below:

You may feel familiar with this when you look at it, yes, it is actually the inspiration for the floating Hallelujah Mountains in the film Avatar, it was formerly called Southern Sky Column and renamed to its current name in 2010 after Avatar became very popular.
The shape of the giant stone pillar is very similar to a skyscraper, it stands 150 meters with no tapering, which is measured from its foot to the peak while the elevation surpassing 1000 meters, which cannot be seen as the height of the pillar itself.