How do I expand $(1+x)^{1/x}$ for small $x$? The binomial expansion
$$(1+x)^{n} = 1 + nx + \frac{n(n-1)}{2}x^{2}+...$$
didn't work because of the $n$ term being undefined at $x=0$.
Taylor expansion doesn't work either since it too would depends on an undefined $1/x$ term.
How does one do it?

Binomial expansion whilst taking $n=1/x$ looks fine to me (though not a power series representation). True, your series is technically not defined at $x=0$ but then neither is the function $(1+x)^{1/x}$, but they have a well defined limit.
By Newton's generalized binomial theorem, for nonzero $x\in (-1,1)$, we have
$$(1+x)^{1/x}=1+nx+\frac{n(n-1)}{2!}x^2+\frac{n(n-1)(n-2)}{3!}x^3+...|_{n=1/x}\\
=1+1+\frac{(1-x)}{2!}+\frac{(1-x)(1-2x)}{3!}+...$$
Note that the limit of the LHS as $x\rightarrow 0$ is $e$, while the RHS (second line) evaluated at $x=0$ is a series expansion of $e$.
