Which binds first, product or factorial?

This would depend on the author, but the former notation would be much more common: $$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$

If the product itself was factorialized, it would most likely be written as the latter: $$\Bigg(\prod_{i \in I}i\Bigg)!$$

edit: added the bolded word much.

The convention \begin{align*} \prod_{i \in I}i! = \prod_{i \in I}(i!)\tag{1} \end{align*} is also affirmed by the operator precedence rules stated in OEIS.

• For standard arithmetic, operator precedence is as follows:

  1. Parenthesization,

  2. Factorial,

  3. Exponentiation,

  4. Multiplication and division,

  5. Addition and subtraction.

and since the product sign $\prod$ is just a short-hand for successively using the multiplication operator, the convention (1) is valid.

I would see it as $$\prod_{i \in I}i! = \prod_{i \in I}(i!)$$

Like the $\sum _i a_i^2$ which is $\sum _i (a_i^2)$ not $(\sum _i a_i)^2$