  Inspired by the work of Schur on the Taylor series of the exponential and
Laguerre polynomials, we study the Galois theory of trimmed exponentials
$f_{n,n+k}=\sum_{i=0}^{k} \frac{x^{i}}{(n+i)!}$ and of the generalized Laguerre
polynomials $L^{(n)}_k$ of degree $k$. We show that if $n$ is chosen uniformly
from $\{1,\ldots, x\}$, then, asymptotically almost surely, for all $k\leq
x^{o(1)}$ the Galois groups of $f_{n,n+k}$ and of $L_{k}^{(n)}$ are the full
symmetric group $S_k$.
