  This manuscript develops a novel understanding of non-polar solutions of the
discrete Painlev\'e I equation (dP1). As the non-autonomous counterpart of an
analytically completely integrable difference equation, this system is endowed
with a rich dynamical structure. In addition, its non-polar solutions, which
grow without bounds as the iteration index $n$ increases, are of particular
relevance to other areas of mathematics. We combine theory and asymptotics with
high-precision numerical simulations to arrive at the following picture: when
extended to include backward iterates, known non-polar solutions of dP1 form a
family of heteroclinic connections between two fixed points at infinity. One of
these solutions, the Freud orbit of orthogonal polynomial theory, is a singular
limit of the other solutions in the family. Near their asymptotic limits, all
solutions converge to the Freud orbit, which follows invariant curves of dP1,
when written as a 3-D autonomous system, and reaches the point at positive
infinity along a center manifold. This description leads to two important
results. First, the Freud orbit tracks sequences of period-1 and 2 points of
the autonomous counterpart of dP1 for large positive and negative values of
$n$, respectively. Second, we identify an elegant method to obtain an
asymptotic expansion of the iterates on the Freud orbit for large positive
values of $n$. The structure of invariant manifolds emerging from this picture
contributes to a deeper understanding of the global analysis of an interesting
class of discrete dynamical systems.
