Is this infinite product always irrational?

Question

Let n be a positive integer$>1$. Is the infinite product $n^{1/n}(n^2)^{1/n^2}(n^4)^{1/n^4}(n^8)^{1/n^8}...$ always irrational?

Answer 1

Almost certainly yes, but it would be extremely surprising if any of these could be proven to be irrational. It's hard enough proving something irrational when you have a closed form formula for it.