Independence of a random variable $X$ from itself

The only events that are independent of themselves are those with probability either $0$ or $1$. That follows from the fact that a number is its own square if and only if it's either $0$ or $1$. The only way a random variable $X$ can be independent of itself is if for every measurable set $A$, either $\Pr(X\in A)=1$ or $\Pr(X\in A)=0$. That happens if and only if $X$ is essentially constant, meaning there is some value $x_0$ such that $\Pr(X=x_0)=1$.