As mentioned by GH from MO, the irrationality measure
for almost all real numbers is 2. However,
computing it for a particular number
is a notoriously difficult problem. For an irrational algebraic number this measure is indeed 2, but this is a pretty hard theorem by Roth for which he got a Fields medal.
Other then that, to my knowledge the only "famous constant" with the known irrationality measure is the number $e$. It is, again, equal to 2 and this is a very old result essentially known to Euler. (It follows from the expansion of $e$ into a continued fraction.) For other "famous", or even not that famous, constants only upper bounds are known.