I recently came across the concept of the irrationality measure. It really fascinated me and when I was looking for known values $\mu(x)$ for mathematical constants $x$, I also came across this paper: On the irrationality measure of arctan 1/3 Unfortunately it could't help me out quite well. What is the irrationality measure of $\arctan(1/3)$? Are there upper and lower bounds? Which famous constant have known irrationality measures or upper/lower bounds? (Except for the ones, that can easily be found on mathworld e.g. $\pi, \ln(2),\ln(3), \pi^2, \zeta(3)$)?
The irrationality measure of $\arctan(1/3)$ is not known. It lies between $2$ and $6.096755\dots$. The lower bound is trivial (it holds for every irrational number), while the upper bound is the main result of the paper you quote. It is reasonable to guess that the irrationality measure of $\arctan(1/3)$ is exactly $2$, because almost all irrational numbers (in Lebesgue measure) have this property.
Regarding your last, rather open ended question, I recommend that you use standard tools such as MathSciNet and Zentralblatt to search the mathematical literature.
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:
$\begingroup$ This looks like a comment to me. $\endgroup$– WojowuApr 8, 2020 at 16:09
$\begingroup$ You are right, but it's too long for a comment. $\endgroup$ Apr 8, 2020 at 16:10
$\begingroup$ I think this is a pretty good answer.I upvoted it. $\endgroup$ Apr 8, 2020 at 20:16
2$\begingroup$ Champernowne's constant 0.123456789101112... is famous though slightly artificial. It has irrationality measure exactly 10. $\endgroup$– dkeApr 8, 2020 at 21:14