Some of the most memorable constants in mathematics like pi and e are transcendental numbers.
Is this true of some or all of the physical constants?
How would you know?
Even the most precisely measured of the physical constants is only known to a dozen or two digits - too few to directly determine whether it was transcendental or rational, or even to make a reasonable guess.
But, if you could come up with a formula from which a physical constant could be determined that had plausible reasons to be correct, perhaps you could know from the form of the formula, even if it wasn't actually possible to calculate the formula numerically to much more precision than the experimental measurement.
Of course, any physical constant with a factor of pi or e in it would be transcendental, regardless of the nature of the remaining factor (except in the modulo unique case where the remaining factor contained the inverse of pi or e as the case might be, for example).
There should be a term for a number that is still transcendental, even after factoring out well defined, purely mathematical constants. Physically transcendental perhaps?