Numbers with known finite irrationality measure greater than 2

Irrationality measure is a question about approximation by rationals. The continued fraction expansion gives the best approximations and controls their quality. Irrationality measure is a kind of asymptotic growth of the continued fraction expansion. Asking about the irrationality measure of a particular number is asking properties of its continued fraction expansion. But if you are willing to specify numbers by their continued fraction expansion, it is easy to write down a continued fraction expansion with the desired measure. Inductively define the continued fraction $[a_1,a_2,\ldots]$ by setting $a_{n+1}=\lceil q_n^{\mu-1}\rceil$, where the convergent is $\frac{p_n}{q_n}$. Then $q_{n+1}=a_{n+1}q_n+q_{n-1}$, so the error $(q_nq_{n+1})^{-1}$ is about $q_n^{-\mu}$.

It's a bit of a cop-out, but it's definitely worth mentioning.

The answer is yes - see for example

Yann Bugeaud Diophantine approximation and Cantor sets Math. Ann. (2008) 341:677–684

The irrationality measure of the Champernowne constant $C_b$ in base $b>2$ is exactly $b$.