Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student:
Find an approximation of $\pi$ formed of the digits $0$ through $9$, each used at most once, combined with (unlimited quantities of) such basic symbols as the elementary arithmetical operations, parentheses, root signs, superscripts indicating exponentiation, and factorial signs.
I don't recall anymore how specific they were about which symbols were allowed, but this is probably close enough. I came up with
$$ \pi = \left(\left( - \frac{1}{2} \right)! \right)^{6/3}, $$
which unfortunately was denied the prize. However, that's not the point of the present question.
I also considered expressions of the form $$ \pi \stackrel{?}{\approx} \sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}} $$ for $n > 2$ not violating the requirements of the puzzle. However, the question I couldn't answer was whether or not suitable combinations of factorial signs and square roots would provide arbitrarily close approximations of $\pi$.
Generalizing this, the question I couldn't answer was this one:
Let $n>2$ be an integer. Let $r_{k,m} = \sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$ with $k$ factorial signs and $m$ square roots. Do the $r_{k,m}$ lie dense in $\mathbb{R}_{>1}$? Does it make a difference which $n$ we start with?