  We consider gravitational lensing of a background source by a finite system
of point-masses. The problem of determining the maximum possible number of
lensed images has been completely resolved in the single-plane setting (where
the point masses all reside in a single lens plane), but this problem remains
open in the multiplane setting. We construct examples of $K$-plane point-mass
gravitational lens ensembles that produce $\prod_{i=1}^K (5g_i-5)$ images of a
single background source, where $g_i$ is the number of point masses in the
$i^\text{th}$ plane. This gives asymptotically (for large $g_i$ with $K$ fixed)
$5^K$ times the minimal number of lensed images. Our construction uses Rhie's
single-plane examples and a structured parameter-rescaling algorithm to produce
preliminary systems of equations with the desired number of solutions.
Utilizing the stability principle from differential topology, we then show that
the preliminary (nonphysical) examples can be perturbed to produce physically
meaningful examples while preserving the number of solutions. We provide
numerical simulations illustrating the result of our construction, including
the positions of lensed images as well as the structure of the critical curves
and caustics. We observe an interesting ``caustic of multiplicity'' phenomenon
that occurs in the nonphysical case and has a noticeable effect on the caustic
structure in the physically meaningful perturbative case.
