  The loop graph of an infinite type surface is an infinite diameter hyperbolic
graph first studied in detail by Juliette Bavard. An important open problem in
the study of infinite type surfaces is to describe the boundary of the loop
graph as a space of geodesic laminations. We approach this problem by
constructing the first examples of 2-filling rays on infinite type surfaces.
Such rays accumulate onto geodesic laminations which are in some sense filling,
but without strong enough properties to correspond to points in the boundary of
the loop graph. We give multiple constructions using both a hands-on
combinatorial approach and an approach using train tracks and automorphisms of
flat surfaces. In addition, our approaches are sufficiently robust to describe
all 2-filling rays with certain other basic properties as well as to produce
uncountably many distinct mapping class group orbits.
