We precisely evaluate the operator norm of the uncentred Hardy-Littlew
ood maximal function on L(p)(R(1)). Consequently, we compute the opera
tor norm of the 'strong' maximal function on L(p)(R(n)), and we observ
e that the operator norm of the uncentred Hardy-Littlewood maximal fun
ction over balls on L(p)(R(n)) grows exponentially as n --> infinity.