Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent. In addition, all of the stated expectations are bounded away from zero. We know that $ E(X)^{-1} \neq E(1/X)$ unless we look at trivial cases.
Is there some way to simplify
$$ E\left\{ \frac{1}{E \left(Y|X \right)} \right\} = \int \left\{ \int y f_{Y|X} \left(y|x \right) \, dy \right\}^{-1} f_{x}(x)dx ?$$