Given a smooth function $f(x):$
Does there always exist an expansion of x around a point $x_{0}$ of the form $$\sum_{n=0}^\infty \frac{h(f^{(n)}(x_{0}))}{n!}g(x-x0,n)$$ for some functions $h(c),g(a,b)$?
As an example, analytic functions have $g(A,n) = A^n$ and $h(y) =y$
No. Let $f(x):=e^{-1/x^2}$ if $x \ne 0$ and $f(0):=0$.
It is well-known that, with $x_0=0$, $f^{(n)}(0)=0$ for $n=0,1,2,...$.
Hence, whatever $g(x-x_0,n)$ is, we have
$\sum_{n=0}^{\infty}{f^{(n)}(x_0)/n!*g(x-x0,n)}=0 \ne f(x)$ for $x \ne 0$.