= Generic superfactorial derivative growth at rational points
Outside a <meagre set> in $C^\infty([0,1])$, every rational $q\in(0,1)$ and every positive integer $M$ admit $m\geq M$ such that
$$
|f^{(m)}(q)|>m!m^m.
$$
For fixed $q,M$, the union over $m\geq M$ of these strict-inequality sets is open. It is dense because a perturbation $\delta\cos(K(x-q)-m\pi/2)$ can make the $m$th derivative arbitrarily large while keeping any prescribed finite collection of lower derivatives arbitrarily small. The <Baire category theorem> applied over the countable pairs $(q,M)$ gives the claim.
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