  Let R_n be the radius of the largest disk covered after n steps of a simple
random walk. We prove that almost surely limsup_{n \to \infty}(log R_n)^2/(log
n log_3 n) = 1/4, where log_3 denotes 3 iterations of the log function. This is
motivated by a question of Erd\H{o}s and Taylor. We also obtain the analogous
result for the Wiener sausage, refining a result of Meyre and Werner.
