Order of growth of derivatives at given x

Question

Is there such an $f$ smooth function and $x\in D_f$, so that the sequence $f(x), f'(x), f''(x), ...$ grows faster than exponential?

Can it grow at a factorial rate or faster?

Answer 1

For a single point $x$, it can be arranged that $f^{(n)}(x)=d_n$ for an arbitrary sequence $d_n$.