jmc

There are two ways for $x$ to not be in the domain of $g$: it can not be in the domain of $f$, or it can be in the domain of $f$ but not in the domain of $f\circ f$. In the first case, the denominator of $f$ is zero, so $$x-3=0\Rightarrow x=3.$$For the second case, we see that the denominator of $f(f(x))$ is $\frac{1}{x-3}-3$. If this is zero, we have $$\frac{1}{x-3}=3⟹x-3=\frac{1}{3}⟹x=3+\frac{1}{3}=\frac{10}{3}.$$This is greater than $3$, so the largest $x$ not in the domain of $g$ is $\boxed{\frac{10}{3}}$.