HeliumMountain

Mountains are not really very tenable objects.  They get thrown up by tectonic plate collisions or volcanism, but gravity and the weather then flatten them pretty quickly.  That is to say, quickly if you are a stone.  The longest-lived mountains tend to be ones that are made of low-density rock, or that float on a low-density area of plate.  Those are more-or-less in equilibrium, like ships on the sea.

This gives a clue how to make an artificial mountain.  You make it from bricks filled with helium.  The bricks would be tough rectangular plastic bags with aluminised interiors to retain the gas.  Each brick might be about the size of a house, and it would come with tabs on the edges to allow it to be attached to all its neighbouring bricks in the same sort of very strong bond pattern that you find in an ordinary brick wall.  Instead of the tabs, you might even be able to do something clever with Velcro on the surfaces.

As the mountain of bricks gets higher, you reduce the pressure of the helium in the bricks to match that of the surrounding atmosphere.  Some reasonably straightforward calculations should allow the entire structure to have neutral buoyancy as a whole, and also at every horizontal slice at all altitudes.  The individual bricks, whilst piled kilometres high, would not be being compressed by those above them, and would be almost completely unstressed.

If you made the helium mountain big, it would probably be a good idea to anchor it to something reasonably substantial in order to resist the wind.  For example, you could build it round an existing natural mountain.

You could then build your helium mountain right up to space.

Once you had done that, you could run a linear accelerator up the side and launch payloads into orbit electrically for a few dollars a kilogram...