One of the interesting consequences of the recent redefinition of the SI units is that now h, k, e and a few others are considered exact constants, just like c. A caveat: physicists are well aware that their trade is about mathematical models of magnitudes they can measure (and extra stuff they can't yet comes in handy to do that), not reality or the physical world in itself, whatever that might be it's something dealt with epoché.
So now the Stefan-Boltzmann constant, which involves powers of c, k and h is exact. It also involves the fifth power of Pi (through the value of the Riemann zeta fuction for argument 4). As you can measure the Stefan-Boltzmann constant being creative with a radiometer, you could determine increasingly accurate values of Pi experimentally. In the sense of magnitudes as used in physics, which is a fruitful approach to being quantitative in this confusing physical world, this "thing" we call "Pi" behaves like a dimensionless physical constant.
Before you'd object, if we came up with more precise theories, experience shows that that would mean corrections to the laws implied above. It also shows that when this happens instances of "Pi" don't deviate from the Platonic value of Pi.