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 \(S_k\).