<s>[INST] The asymptotic expansion of $x^x$ at $0$. If we put $x^x$ in Wlofram Alpha, we get the following:
$$
x^x = 1 + x \log(x) + \frac12 x^2 \log^2(x) + \frac16 x^3 \log^3(x) + \frac1{24} x^4 \log^4(x) + \frac1{120} x^5 \log^5(x) + O(x^6)
$$
Is there a name of this form of expansion and is there a systematic way to find such expansions?
 [/INST] We know that 
$$x^x=e^{x\log x}$$
Also by Taylor's Series,
$$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots$$
So
$$\begin{align}
x^x&=e^{x\log x}\\
&=1+x\log x+\frac{(x\log x)^2}{2!}+\frac{(x\log x)^3}{3!}+\frac{(x\log x)^4}{4!}+\frac{(x\log x)^5}{5!}+\cdots\\
&=1+x\log x+\frac{1}{2}x^2\log^2 x+\frac{1}{6}x^3\log^3x+\frac{1}{24}x^4\log^4x+\frac{1}{120}x^5\log^5x+O(x^6)
\end{align}$$
</s>