Suppose you a function $f$ which is differentiable,
with the property that
$$ f^{(n)} (0) = (n!)^2 $$
And in general
$$ f^{(n)} (a) = O((n!)^2)$$
For any $a \in \mathbb{R}$.
This function then is nowhere well defined by a taylor series. But I don't immediately accept that this function "doesn't exist". What if we have some degenerate super fast growing function (like an analytically continued nested ackermann or something along those lines).
Can anyone give an example of a function that satisfies the above equations, or generally a function such that
$$ f^{(n)}(a) = 2^{O(\Omega(n \ln (n) ))} $$
There is no holomorphic function (on any neighborhood of $0$) with $f^{(n)}(0) = (n!)^2$, but you can find a $C^\infty$ smooth function with arbitrarily prescribed partial derivatives (this is a theorem of Borel). In the complex setup, you can for example ask for a smooth function satisfying $$ \frac{\partial^n f}{\partial z^n}(0) = (n!)^2 \qquad \frac{\partial^{k+l} f}{\partial \bar z^k \partial z^l}(0) = 0 $$ for all $l$ and $k \ge 1$