I would have expected more numbers originating from physics, like Reynolds number (bad example since it is not really constant though).

The human-invented ones seem to be just a grasp of dozens man can come up with.

i to the power of i is one I never heard of but is fascinating though!

To prove something is transcendental we would need to know how to compute it exactly, and I’m struggling to see how that would come up frequently in a physics context. In physics most constants are not arbitrary real numbers derived from a formula, they’re a measured relationship, which sort of inherently can’t be proved to be transcendental

some constants that may or may not be transcendental: - Percolation Thresholds: https://en.wikipedia.org/wiki/Percolation_threshold - Critical scalings in 3d: https://en.wikipedia.org/wiki/Universality_class#Ising_model

Yeah I'd expect Bessel function zeroes and such