  Let $R$ be a ring of characteristic $0$ with field of fractions $K$, and let
$m\ge2$. The B\"ottcher coordinate of a power series $\varphi(x)\in x^m +
x^{m+1}R[\![x]\!]$ is the unique power series $f_\varphi(x)\in x+x^2K[\![x]\!]$
satisfying $\varphi\circ f_\varphi(x) = f_\varphi(x^m)$. In this paper we study
the integrality properties of the coefficients of $f_\varphi(x)$, partly for
their intrinsic interest and partly for potential applications to $p$-adic
dynamics. Results include: (1) If $p$ is prime and $R=\mathbb Z_p$ and
$\varphi(x)\in x^p + px^{p+1}R[\![x]\!]$, then $f_\varphi(x)\in R[\![x]\!]$.
(2) If $\varphi(x)\in x^m + mx^{m+1}R[\![x]\!]$, then
$f_\varphi(x)=x\sum_{k=0}^\infty a_kx^k/k!$ with all $a_k\in R$. (3) In (2), if
$m=p^2$, then $a_k\equiv-1\pmod{p}$ for all $k$ that are powers of $p$.
