Define $F(n,i)=\prod_{j=1}^nj^{j^i}$. $F(n,0)=n!$ is just the standard factorial, whereas $F(n,1)$ is the so-called hyperfactorial.

Is there a term for $F(n,i)$?

How fast do these grow?

Is the growth rate given by $2^{\frac{c_in^{i+1}\log_2n}{i+1}}$ with some constant $c_i>0$ (based on comments below)?

Is there a relation to some special function for every $i$ (just like we have for the factorial and the hyperfactorial)?