Repeated Factorials and Repeated Square Rooting

I was talking with friends about silly questions involving what numbers you can get using only a single digit "3" and unary operations. We eventually conjectured that using only factorials and square roots you can get arbitrarily close to any number greater than or equal to $1$. But we are having trouble proving or disproving the conjecture.

Precisely, start with the number $3$. Then take its factorial $m$ times, and then take the square root of its result $n$ times. Ie, $(3!!\ldots !)^{\frac{1}{2^n}}$ where there are $m$ factorials. Let the set of numbers achievable in this way be $X$. Is $X$ dense in $[1,\infty)$?

The only progress we have made is to show that any interval $[x,x^2]$ for $x>1$ there is a limit point $a \in [x,x^2)$ of $X$. This is true because for any $z > x^2$ we can square root it an appropriate number of times to get it in $[x,x^2)$. We can do this for infinitely many points of the sequence $3,3!,3!!,\ldots$. And all points we get through this process are distinct. Let $F(m)$ be $3$ with $m$ factorials. If $F(m)^{\frac{1}{2^n}} = F(a)^{\frac{1}{2^b}}$ then we can raise each side to a power and get an expression of the form $F(m) = F(a)^{\frac{1}{2^c}}$. But factorials are not squares. (To see this for $q!$, note that there is a prime in $[q/2,q]$ by bertrand's postulate. This prime appears only once in the factorization of $q!$). So all numbers we get are distinct. We have infinitely many distinct points in $X \cap [x,x^2)$ and therefore there is a limit point.

Other than that, we can't figure anything out. It feels like $X$ should be dense. Consider some interval $[x,x^2]$. Take lots of factorials of $3$. Then take square roots of that until it falls in $[x,x^2]$. It feels like the points we get will be somewhat uniformly distributed around $[x,x^2]$ and therefore dense.

Solution 1:

A related conjecture is posted at this link-that with the floor function you can get all naturals. I remember seeing a proof to that version with $\pi$ instead of $3$ a while ago, but can't find it now.