Backing out some numbers from the piece, the 2x2 held 950 lbs, which is roughly 1000 lbs in 1/2"x1/2", or 4000psi. That's roughly the strength of ordinary unreinforced concrete, at a far lighter density. Legos are also similar to unreinforced concrete in their tensile strength, which is low, variable, and brittle. The usual calculation is that tensile strength is ~ 10% of the compressive strength for concrete, but it depends greatly on the cracks and other discontinuities.

From the problem set though, there's an interesting effect. If you taper the tower with an exponential curve, 1/e^x, the pressure on the bottom of the tower can be constant as you increase the both the footprint and the height. Not coincidentally, that's the same curve you find in towers in the real world like the Eiffel Tower and the CN Tower.

The ultimate height that you could make with a tower would certainly depend on what constraints you're applying. Is there a limited number of bricks? A limited base area? Any supports at all? How do the people actually assemble the thing? What safety regs are there?

With no constraints, I don't see a reason that legos couldn't be built to the height of the great pyramids. Apart from the obvious one that it would be hellaciously expensive.

Once you start talking about constraints and something more tower shaped than mountain shaped, stability is the biggest concern. Elastic stability will affect the tower, at least as an upper limit to the height/cross section ratio.

So why can't we build a space elevator?

Earth's radius is 4k miles. Geostationary orbit is ~25k. Space elevators IIRC, are proposed for ca 60k miles out. Assuming a 10-20x height/width ratio, the base would be 3-6k miles on a side. Even just getting to geostationary orbit would require a mountain with a base the size of a continent.

And for extra credit, the tower should survive a run on the shake table.