How High Can Humans Build?

Courtesy of Nathan S.:

I've pasted the link to a vsauce video: "How High Can We Build". This topic was interesting to me for its relevancy, as Earth's ever expanding population will eventually force us to somehow use space more efficiently, building into the sky would certainly help in this endeavor. The topic also interested me as the reason we have a height limit is due to gravity which we study on a regular basis.