  Let $(X,\mathcal{B}, \mu, T)$ be an ergodic dynamical system on a non-atomic
finite measure space. Consider the maximal function $\dis R^*:(f, g) \in
L^1\times L^1 \to R^*(f, g)(x) = \sup_{n} \frac{f(T^nx)g(T^{2n}x)}{n}.$ We show
that there exist $f$ and $g$ such that $R^*(f, g)(x)$ is not finite almost
everywhere. Two consequences are derived. The bilinear Hardy--Littlewood
maximal function fails to be a.e. finite for all functions $(f, g)\in L^1\times
L^1.$ The Furstenberg averages do not converge for all pairs of $(L^{1},L^{1})$
functions, while by a result of J. Bourgain these averages converge for all
pairs of $(L^{p},L^{q})$ functions with $\frac{1}{p}+\frac{1}{q}\leq 1.$
