Suppose you have a finite product of recursive functions (e.g. primitive recursive). The result function will be primitive recursive.
But what if we count the infinite product of primitive recursive functions. Will the product be recursive or non-recursive?
Generally, any infinitary operation on functions need not preserve recursiveness.
In this particular case, let $f_i(x)=1$ if $x$ did not enter the halting problem at stage $i$ exactly, and let $f_i(x)=2$ if $x$ has entered the halting problem at stage $i$ exactly. Then the $f_i$s are (even uniformly) recursive, but the infinite product $$p:x\mapsto\prod_{i\in\mathbb{N}}f_i(x)$$ is Turing-equivalent to the halting problem: $x$ is in the halting problem iff $p(x)=2$.