Let \begin{equation} z := \prod_p p^{1/p^2}, \end{equation}

where the product is over all prime numbers $p$, and we always take the positive real root. Is $z$ transcendental or algebraic, or (as I suspect) is the answer not at all clear (meaning it's "probably" transcendental)?

More generally, let $(p_i)$ be an increasing sequence of prime numbers, and $(e_i)$ a sequence of integers such that the infinite product $\prod_{i=1}^\infty (p_i)^{1/e_i}$ converges.

Are there some numbers of this form where we can say anything, other than those where the convergence is so fast that we can use classical Liouville-type arguments?

Edit: Just to clarify: I'm not sure if there's any natural reason to look at such numbers, except that I had a vague idea (which is too long to get into here) that I might be able to prove something about them. I guess that was the case in the beginning of transcendence theory (a subject I don't usually think about) -- the first numbers to be proved transcendental were not numbers anyone cared about for any other reason.

Another comment about these numbers: they "look" transcendental, because the partial products live in larger and larger extensions of the rationals. However, a power of each partial product is an integer, so in some sense we are not getting too far away from the rationals -- or away from $\mathbb{Q}^\times \otimes \mathbb{Q}$.