This is a follow-on of sorts to this question, but is self-contained.
Let $F_1 := \{f \in C^\infty(\mathbb{R}) \mid \|\frac{df}{dx}\|_\infty \le c_1\}$ ($c_i > 0$ throughout).
Given $f, g \in F_1$, we have by the chain rule that $g \circ f \in F_1$ if
$\|\frac{dg}{df}\|_\infty \|\frac{df}{dx}\|_\infty \le c_1$
and this is automatically true if $c_1 \le 1$.
Now let $F_2 := \{f \in F_1 \mid \|\frac{d^2f}{dx^2}\|_\infty \le c_2\}$.
Given $f, g \in F_2$, we have by the generalized chain rule (Faà di Bruno's formula) that $g \circ f \in F_2$ if it is in $F_1$ and
$\|\frac{d^2g}{df^2}\|_\infty \|\frac{df}{dx}\|_\infty^2 + \|\frac{dg}{df}\|_\infty \|\frac{d^2f}{dx^2}\|_\infty \le c_2$
and this is automatically true if $c_1, c_2$ satisfy the corresponding inequality $c_2 c_1^2 + c_1 c_2 \le c_2$, i.e. if $c_1^2 + c_1 \le 1$ (in which case $c_1 \le 1$ also).
In general, let $F_n := \{f \in F_{n-1} \mid \|f^{(n)}\|_\infty \le c_{n} \}$.
Given $f, g \in F_n$, we have by the generalized chain rule that $g \circ f \in F_n$ if it is in $F_{n-1}$ and
$\sum_{\pi\in\Pi(n)} \|g^{(|\pi|)}\|_\infty \prod_{B\in\pi} \|f^{(|B|)}\|_\infty \le c_n$ (notation from the Wikipedia article)
and this is automatically true if the $c_i$ satisfy the corresponding inequalities
$\sum_{\pi \in \Pi(i)} c_{(|\pi|)} \prod_{B \in \pi} c_{(|B|)} \le c_i \forall i \in \{1, ..., n\}$.
My question then is: can $F_\infty := \bigcap_{n \in \mathbb{N}} F_n$ be closed under composition without being trivial? In other words:
Does there exist a sequence of values $c_i > 0$ which satisfy the inequalities
$\sum_{\pi \in \Pi(i)} c_{(|\pi|)} \prod_{B \in \pi} c_{(|B|)} \le c_i \forall i \in \mathbb{N}$?