An series $\{a_n\}$ to a function $f(x)$ is defined as
$$ f(x) - \sum\limits_{n=0}^{N} a_n x^n\sim a_{N+1}x^{N+1} $$
as $x \rightarrow x_0$ for all N.
I have just heard, that the exponents $n$ do not have to be integers, but fractional values for example are allowed too.
What about the $n$ being negative, are Laurent series potentially valid asymptotic series approximating some functions too?
If not, why not?
For some functions, one can find asymptotic series that have nothing to do with powers of $x$ at all. So negative powers, fractional powers, all are possible. In general, asymptotic series expansions for a function $f(x)$ have the form
$$ f(x) \sim \sum_{n=1}^{\infty} f_n(x). $$
See Wikipedia for some examples of such asymptotic series.