Suppose we have a random variable $X: \Omega \to \mathcal{X}$, with pdf $f_X$. It is clear that the expectation of any function of $X$, say $g(X)$, is
$$E[g(X)] = \int_{x \in \mathcal{X}}g(x)f_X(x)dx$$
Suppose we wish to take the expectation of the random variable $f_X(X)$, (in other words, what value the pdf will take in expectation), such that
$$E[f_X(X)] = \int_{x \in \mathcal{X}}f_X(x)f_X(x)dx = \int_{x \in \mathcal{X}}f^2_X(x)dx$$
Does this expectation have any meaningful interpretation?