As a reference for anyone encountering this discussion, I thought I'd mention Natural Units explicitly. Basically, they are the systems of units that particle physicists use. They are attempts to normalize out as many fundamental constants as possible, exactly as you discuss.
Unfortunately, you can't build a system that gets them all. You are always left with some fraction over pi, or the square root of the fine-structure constant, or something.
In the post Complexity and Intelligence, Eliezer says that the Kolmogorov Complexity (length of shortest equivalent computer program) of the laws of physics is about 500 bits:
Where did this 500 come from?
I googled around for estimates on the Kolmogorov Complexity of the laws of physics, but didn't find anything. Certainly nothing as concrete as 500.
I asked about it on the physics stack exchange, but haven't received any answers as of yet.
I considered estimating it myself, but doing that well involves significant time investment. I'd need to learn the standard model well enough to write a computer program that simulated it (however inefficiently or intractably, it's the program length that matters not it's time or memory performance).
Based on my experience programming, I'm sure it wouldn't take a million bits. Probably less than ten thousand. The demo scene does some pretty amazing things with 4096 bits. But 500 sounds like a teeny tiny amount to mention off hand for fitting the constants, the forces, the particles, and the mathematical framework for doing things like differential equations. The fundamental constants alone are going to consume ~20-30 bits each.
Does anyone have a reference, or even a more worked-through example of an estimate?