While I have no doubt it would require an unreasonable amount of material and resources, so not doable... I wonder.... would a space-elevator made out of modern materials, but shapped like a pyramid with a broad base, be able to be made tall enough to extend into space. I figure if the base is wide enough and the materials at least as strong as solid rock, it should be possible... but without running the numbers I dunno... I do know there is a limit on the maximum size of a mountain so .... i dunno.
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@freemo If I gather correctly, you are thinking of a construction that is loaded ~only in compression. Am I right?
What's the order of magnitude of the size you're thinking about (it's not obvious, because it depends on the angle in the pyramid's tip)? Less than 1% of Earth's mass/more than that?