For a given real number $x$, let $R_x$ be the set of real numbers $r$ such that the inequality

$$\displaystyle \left| x - \frac{p}{q} \right| < \frac{1}{q^r}$$ has at most finitely many solutions with integers $p,q$. Define the irrationality measure of $x$, say $\mu(x)$, to be the infimum of $R_x$.

It is known that if $x$ is algebraic and not rational, then $\mu(x)$ is 2, by Roth's Theorem. It is trivial that if $x$ is rational, then $\mu(x) = 1$. I believe it is also known that all real numbers except a set of measure 0 has irrationality measure of 2, but I am unsure of the reference.

For some known transcendental numbers, upper bounds for $\mu$ are known. For example, we know that $\mu(\pi) < 7.6063$ (Salikhov, V. Kh. "On the Irrationality Measure of ." Usp. Mat. Nauk 63, 163-164, 2008. English transl. in Russ. Math. Surv 63, 570-572, 2008.)

Are there any general results concerning a set of transcendental numbers $x$ with $\mu(x) = 2$? Are there any known, 'interesting' numbers (expressible in well-known functions or constants) $x$ with $\mu(x) = 2$?

2more comments