One of the goals of modular representation theory is to understand projective resolutions of finitely generated modules for a modular group algebra. Eisenbud has proved that a module whose minimal resolution is bounded in dimension is periodic, in the sense that the modules and maps in the minimal resolution repeat with a certain period. The period is bounded by the largest degree of a non-nilpotent generator of cohomology. If \(G\) is a \(p\)-group and \(K\) is a field of characteristic \(p\), the period of an indecomposable periodic \(KG\)-module is always of the form \(p^ a\) if \(p = 2\), and \(p^ a\) or \(2p^ a\) if \(p\) is odd (see Proposition 2.2). In this paper, we prove that for \(p = 2\) the integer \(a\) may be chosen arbitrarily large, by choosing the group and the module appropriately. Theorem 1.1. Given a positive integer \(a\), and a field \(K\) of characteristic 2, there exists a finite 2-group \(G\) and a periodic \(KG\)-module \(M\) whose period is exactly \(2^ a\). The groups \(G\) in question may be taken to be extraspecial 2-groups of width \(2a\).