Why the maximum height of a mountain on earth is ~10 km.

It can be provided by considering the elastic properties of rocks.

A mountain base is not under uniform compression and this provides some shearing stress to the rocks under which they can flow. The stress due to all the material on the top should be less than the critical shearing stress at which the rocks flow.

At the bottom of a mountain of height h, the force per unit area due to the weight of the mountain is h ρ g where ρ is the density of the material of the mountain and g is the acceleration due to gravity.

The material at the bottom experiences this force in the vertical direction, and the sides of the mountain are free. Now the elastic limit for a typical rock is 30 × 10³ N m^-2.

On equating, we get h ρ g = 30 × 10³ N m^-2

On solving, we get h = 10 km