Summary: Let \(S_0,\dots,S_n\) be a symmetric random walk that starts at the origin \((S_0=0)\) and takes steps uniformly distributed on \([-1,+1]\). We study the large-\(n\) behavior of the expected maximum excursion and prove the estimate \[ E\max_{0\leq k\leq n}S_k= \sqrt{{2n \over 3\pi}}-c+{1 \over 5}\sqrt {{2\over 3\pi}}n^{-1/2} +O(n^{-3/2}), \] where \(c=0.297952\dots\). 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, \({n\over 4}+{1\over 2} E\max_{0\leq j\leq n}S_j+ {1\over 2}= {n\over 4}+O (n^{1/2})\), when the rectangle sides are \(2n\) independent uniform random draws from \([0,1]\).