The complex numbers (well, at least those with a rational imaginary part and a rational real part) have been recently proven to be necessary to describe the universe[0] (assuming quantum theory is correct).
The irrational numbers are then are the only numbers that are harder to pin down, and I'm not sure that there is a way to prove that any physical quantity has an irrational value, vs a rational value that is arbitrarily close to that irrational value.
