The Puiseux series expands a variable $x$ into fractional powers with bounded denominators. You tend to get these when looking at functions of the form $f(x^{1/n})$, where $f(x)$ is analytic.
We can drop the restriction that the denominators must be bounded to get the field of Levi-Civita series in $x$, which instead has the restriction that given any rational power $q'$, there are only finitely many terms $x^q$ in the series with $q < q'$. This can be thought of as a metric completion of the field of Puiseux series.
When do these series naturally arise when looking at functions $\Bbb R \to \Bbb R$? That is, are there functions do not have a Puiseux expansion, but which do have a Levi-Civita expansion?