Let $\varrho$ be an irrational number, with irrationality measure $\mu(\varrho)$. I define the function f as
$$f(\varrho)=\liminf_{p,q\in\mathbb{Z},q\to\infty}\left(q^{\mu(\varrho)}\left|\varrho-\frac{p}{q}\right|\right)$$
So $f(\varrho)$ is the smallest number so that $\left|\varrho-\frac{p}{q}\right|<\frac{f(\varrho)}{q^{\mu(\varrho)}}$ has infinitely many solutions.
If $\mu(\varrho)=2$ then by Hurwitz's theorem $f(\varrho)\leq\frac{1}{\sqrt{5}}$ and this limit is sharp because $f\left(\frac{1+\sqrt{5}}{2}\right)=\frac{1}{\sqrt{5}}$.
Are there any other numbers for which f is known? For example, $\sqrt{2}$, $\pi$, $e$, etc.