Summary: For a simple symmetric random walk on the lattice \(\mathbb{Z}^ d\), let \(S_ n = X_ 1 + \cdots + X_ n\) and let \(X_ 1, X_ 2, \dots\) be a sequence of independent and identically distributed random vectors with \[ {\mathbf P}\{X_ i = e_ i\} = {\mathbf P}\{X_ i = -e_ i\} = {1\over 2d}\quad (i = 1,2, \dots, d), \] where \(e_ 1,e_ 2, \dots, e_ d\) are the orthogonal unit vectors of \(\mathbb{Z}^ d\). Denote by \(R_ d(n)\) the radius of the largest ball \(\{x \in \mathbb{Z}^ d: \| x \| \leq r\}\) every point of which is visited at least once in time \(n\). The present paper studies the limiting behavior of \(R_ d(n)\) for \(d = 1\), \(d = 2\), and \(d \geq 3\).