  Fix a prime $p$ and polynomials $f,g\in \mathbb{Q}_p[x]$. If the Newton
polygon of $g$ consists of a single segment with negative slope, we show under
some mild conditions that the Newton polygon of $f\circ g$ is identical to that
of $f$, but stretched horizontally by a factor of $\operatorname{deg} g$.
Specializing to the case $f=g$, this implies that all iterates of certain pure
polynomials are irreducible, recovering a classical result of Odoni on the
dynamical irreducibility of Eisenstein polynomials. We also prove that the
Taylor polynomials of the exponential function (a well-known family of
irreducible polynomials) remain irreducible upon composition with all iterates
of certain pure polynomials of large enough degree.
