What we know
The simple answerWe have too much information for a good dimensional analysis (too many ways of combining our quantities to get the right units). But there does seem to be a natural, straightforward way to do it.
I wonder if
The right answerIf we're moving away from the surface of the Earth, we have to respect our knowledge that gravity goes as
The radius of the earth doesn't really directly affect the orbit. In fact, I used it only because I know it, and I don't know the mass of the Earth or the gravitational constant
This is a pain to calculate mentally.
The computer claims it's 43Mm, which still seems way too high, so what's up?
Checking wikipedia, it turns out that's what's up is the geostationary orbit, which really is 42Mm above the center of the Earth (or 36Mm above your head). Which is wild, but score one at least for dimensional analysis.