I've been playing for a while with the notion of a lunar space elevator. Nothing too heavy, just a Mathematica notebook filled with integrations, 1D graphs and the like.

I think it would be a good place as a proof of concept, less cable stress required and not on Earth for those who would require convincing. It would have to be 2.5 times longer than Brad Edwards concept of an Earth based SE, about 278,000 km(!).

It could be caught at the apogee of an earth orbit. A good way to get down to the moon without a lot of reaction mass. It's easier to leave the moon without reaction mass via a EM launcher or something like that, but could be used for that as well.