Given a $f \in C^\infty_0(\mathbb{R}^n)$ (the set of smooth functions that vanishes at infinity) is in general false that its derivative are bounded, also if the function is bounded (see e.g. $f(x)=\frac{\sin(e^x-1)}{x}$).
For polynomial we know that $P_n^{(n)}(x)=Cn!$ where $C$ is a constant. There is some similar estimation for some subclass of $C^\infty_0$ functions i.e. is it possible to characterize in some way the functions such that for any $n > N_0$ $$ \sup_{x \in \mathbb{R}^n}f^{(n)}(x) \le n! M^n $$ where $N_0$ and $M$ are constants?
EDIT:
the function $ f(x):= e^{-|x|^2} $ satisfies all the conditions so not only the polynomial satisfies this inequality. Is it true also for all the compactly supported functions on $\mathbb{R}^n$?