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?