You'd want tensile strength rather than compressive. The trick is to anchor a counterweight out beyond your target distance and let it pull the weight of the cable up rather than building a tower. Think of swinging a ball on a string rather than building a skyscraper. Assuming a sufficiently sized counterweight you can support the weight of the anchoring cable plus whatever else you want to hang off of it (space dock, elevator terminus, etc.)

stacked tall enough...? The space elevator concept is a geostationary node orbiting earth directly above a fixed point, with cables running between them. Not a gigantic skyscraper up into the sky. What am I missing?

If you build it tall enough, centrifugal force will start pulling on it. Building it that way though... But yeah. Doubt the right material exist atm.

Goats! For the love of God, stay away from the spiders!