Let $R$ be a non-zero commutative ring. We know that any $R$-module $M$ can be written as a quotient of a free module, e.g. $R^{(M)} \to M \to 0$ via $e_m \mapsto m$.
But is there an example of an $R$-module $M$ which is not a quotient of the product $R^J$ for any set $J$ ? This can't work if $M$ is finitely generated, of course (take $J$ to be any set of generators). I guess there should be an example : in general it is difficult to construct morphisms from a product to a given module (but easier from a coproduct to a given module).