We consider a symmetric random walk of length $n$ that starts at the origin and takes steps uniformly distributed on the real interval $[-1,+1]$. We study the large-$n$ behavior of the expected maximum excursion and prove a very precise estimate. This estimate applies to the problem of packing $n$ rectangles into a unit-width strip; in particular, it makes much more precise the known upper bound on the expected minimum height when the rectangle sides are $2n$ independent uniform random draws from $[0,1]$.