Even then there are some dimensionless numbers that you can't get rid of (the ratio between the strength of the gravitational and coulomb forces being one obvious example).

Pi being another.

I wouldn't have said Pi was a physical constant - it's not like we actually measure circles to derive it empirically.

True, but it certainly is a dimensionless number that you can't get rid of (and that actually shows up in physical formulae, such as http://en.wikipedia.org/wiki/Coulomb%27s_law). Come to think of it, so is 2.

I guess if you tried to measure π by constructing circles, you'd actually be measuring the curvature tensor of space (which is an experimental observation), not π (which is just π).

π being, simply, the measurement of the curvature tensor of an ideal Euclidean plane. If we lived in a universe that was non-Euclidean at macro-scale, π would be just another irrational number, and some other quantity would be exalted as "fundamental."

Pi would still be fundamental in mathematics whatever the geometry of our universe. (Examples of where it would turn up: consider the differential equation f''=-f; all its solutions are periodic with period 2pi. The series 1-1/3+1/5-1/7... has sum pi/4. The series 1+1/4+1/9+1/16+... has sum pi^2/6. exp(pi sqrt(163)) is ridiculously close to being an integer. There are deep reasons for all these things, and they wouldn't go away if the universe were very far from spatially flat.)

Natural units are nice, but they are completely impractical for dealing with normal Mirkowskian space. I.e., how do I explain to someone who cannot view me how tall I am? Our normal unit system is nice because it's phenomenological, and maybe inconvenient.

Agreed, but the universe is under no obligation to express itself in units that are convenient on a human scale. I like to think of this as an anti-anthropomorphic principle.