  A few years ago, Bourgain proved that the centered Hardy-Littlewood maximal
function for the cube has dimension free $L^p$-bounds for $p>1$. We extend his
result to products of Euclidean balls of different dimensions. In addition, we
provide dimension free $L^p$-bounds for the maximal function associated to
products of Euclidean spheres for $p > \frac{N}{N-1}$ and $N \ge 3$, where
$N-1$ is the lowest occurring dimension of a single sphere. The aforementioned
result is obtained from the latter one by applying the method of rotations from
Stein's pioneering work on the spherical maximal function.
