Is it possible mathematically for photons, which behave as individual Gaussian wave packets, to combine in such a way that the approximate result is a plane wave at one particular frequency (i.e., the classical plane wave solution to Maxwell's wave equations)?
This isn't strictly true. Photons aren't restricted to being represented by a gaussian wave packet, or really any type of wave packet allowed by Maxwell's equations. A plane wave of one particular frequency is a perfectly valid photon (except that it would occupy an infinite amount of space). Maxwell's equations are the equivalent of the Schrödinger equation for light, and the electric and magnetic fields are the equivalent of the wavefunction. Any solution to Maxwell's equations is a valid photon wavefunction.
You don't need a superposition of gaussian wave packets, the classical plane wave solution is fine on it's own. It is, however, possible to express the classical plane wave solution as a superposition of suitable wave packets.