set of values taken by the density of the random variable $X$ is continuous random variable. $f_X(x)$ is density of $X$, $x\in\mathbb{R}$.
What are possible values of expression $f_X(x)$  ?  

To be more precisely, the question is about image: $f_X(\mathbb{R})$.  

$X$ is not normal, we don't anythhing about it. The only thing we know is continuity, so $P(x\in [a, b])=\int_{a}^{b}f_X(x)dx$
Well, if that is all you know then this is a very general question.

You can easily look up the properties of probability densities on Wikipedia https://en.wikipedia.org/wiki/Probability_density_function.

Densities can take any non negative real value. That and the fact that the integral over the whole real line should be one is all you can say without more information.
If I'm right in recalling that the density of an impulse is a zero-width spike of infinite height, then the value in general, i.e. across all possible density functions, can be any non-zero real number.