What is the mutual information $I(X;X)$?

For any r.v. $X$

$$I(X,X)=H(X).$$

To see this, put $Y=X$ in $I(X,Y)=H(X)-H(X|Y)$ and use

$$H(X|X)=0.~~ (*)$$

In summary, the mutual information of $X$ with itself is just its self information $H(X)$, as $H(X|X)$, i.e. the residual average information carried by $X$ conditional $X$ is zero, i.e. $(*)$ holds.