I'm not sure if it's correct to say "more and more higher" because "more and more expensive" is possible. But as you can see, in the second example the comparative form of the adjective expensive is "more expensive", so this expression indicates that somehting is more expensive each time. Can I use "more and more" with the same usage with short adjectives such as high?
Where the adjective is short enough to take the comparative in -er, as with high, the usual form of the expression is “higher and higher.”
I would say:
That building is getting higher and higher.
I think higher is better, because the building elevation (as in skyscrapers) is probably in such a context. But in the end it is context dependent.
In the context of the question as I understand it, I would rather say:
That building keeps getting higher. (or taller if you prefer)