In the present article we consider multiparameter maximal averages and discover the crucial roles played by the number of parameters in their boundedness properties. The problem we shall deal with is initiated by Rubio de Francia [8] and will be in the spirit of an inductive extension to multiparameter cases, in which tools of our study rely on the theory of Harmonic Analysis on product spaces. Suppose that $d_mu$ is a complex Borel measure supported on a compact subset S of $R^N$ having total mass one, $smallint_S d_mu = 1$.