On $\mathbb{R}$, let $F$ be the constant function $1, G = \frac{d\delta}{dx}$, and $H = \chi_{(0,\infty)}$. Then $(F*G)*H$ and $F*(G*H)$ are well defined in $S'$ but are unequal.
Without doing anything formal, I can intuitively see that:
$G*H = \delta'* \chi_{(0,\infty)} = (\delta * \chi_{(0,\infty)})' = \chi_{(0,\infty)}' = \delta$
and that:
$F*G = (1*\delta') = (1*\delta)' = <1,\delta>' = \left( \int_\mathbb{R} \delta \right)' = 1' = 0$
However, that's intuition and I'm not really using the theory of distributions which I'm struggling with. For example, how do I use the fact that for $F \in D'(U)$ and $\psi \in C_c^\infty(U)$ that $(F*\psi)(x) = <F, \tau_x\tilde{\psi}>$ to show the above (here, $\tau$ is the translation operator and $\psi(-x) = \tilde{\psi}(x)$. In other words, I understand how to intuitively see what the answer should be, but not how to use distribution theory to formally prove it.