  It is well-known that estimates for maximal operators and questions of
pointwise convergence are strongly connected. In recent years, convergence
properties of so-called `non-conventional ergodic averages' have been studied
by a number of authors, including Assani, Austin, Host, Kra, Tao, and so on. In
particular, much is known regarding convergence in $L^2$ of these averages, but
little is known about pointwise convergence. In this spirit, we consider the
pointwise convergence of a particular ergodic average and study the
corresponding maximal trilinear operator (over $\mathbb{R}$, thanks to a
transference principle). Lacey and Demeter, Tao, and Thiele have studied
maximal multilinear operators previously; however, the maximal operator we
develop has a novel bi-parameter structure which has not been previously
encountered and cannot be estimated using their techniques. We will carve this
bi-parameter maximal multilinear operator using a certain Taylor series and
produce non-trivial H\"{o}lder-type estimates for one of the two "main" terms
by treating it as a singular integrals whose symbol's singular set is similar
to that of the Biest operator studied by Muscalu, Tao, and Thiele.
