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.

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.

[EDIT] There is actually a pretty good answer here on MO which I missed somehow:

Numbers with known irrationality measures?