Commercial broadcast, military, governments (diplomatic services?) demining, (ground penetrating radar). cooking, baking, soldering? Diathermy?
Perhaps there are special licenses that allow use of more than the usual power limit for certain purposes like bouncing a signal off the moon?
To do that one doesn't need the legal limit, just a high gain antenna and a bit of power (enough to exceed the noise level at the receiver which presumably is also using a highly directional antenna).
AS I understand it its getting easier and easier to do, affordably and legally. Especially with digital modes.
I have some small directional antennas that could be combined with external dishes (As a feed) to give the requisite gain. Wonder if I could receive hams working EME?
When I lived elsewhere, at a restaurant equipment shop I found chrome plated steel pot lid thats a perfect parabola (as shown by its ability to focus sunlight into a point with absolute precision. It makes my eyes hurt just to think about it. )
Anyway, this does not do much at low microwave frequencies (I suspect for wifi it might deliver ~20 db gain or so.) but the higher you go the more gain you get. This is roughtly an 18 inch dish.. A 1 meter dish is fairly affordable and easy to find. These are popular for direct-TV like services.
Hige parabolic dishes are very impractical because of wind loading in storms. But smaller ones would work for EME on higher bands.