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 $f(f)$. In the first case, the denominator of $f$ is zero, so

$$2x-5=0\Rightarrow x=\frac{5}{2}.$$

For the second case, we see that the denominator of $f(f(x))$ is $\frac{2}{2x-5}-5$. If this is zero, we have

$$5(2x-5)=2\Rightarrow 10x=27\Rightarrow x=\frac{27}{10}.$$

This is greater than $\frac{5}{2}$. So the largest $x$ not in the domain of $g$ is $\boxed{\frac{27}{10}}$.