Mathematica gives the result:
$$x+x^2\ln(x)+\frac{1}{2}x^3(\ln(x)+\ln(x)^2)+\frac{1}{6}x^4(\ln(x)+3\ln(x)^2+\ln(x)^3)+...$$
I have searched for solved examples or step-by-step tutorials calculating Puiseux expansions of functions, but didn't find good sources.
I want to know the formula or the algorithm to obtain such terms up to degree $n$.
You can use the exponential generating function of the Touchard polynomials to obtain $$ x^{{\rm e}^x } = {\rm e}^{{\rm e}^x \log x} = x{\rm e}^{({\rm e}^x - 1)\log x} = x\sum_{n = 0}^\infty \frac{T_n (\log x)}{n!}x^n $$ for $x>0$.