A hollow sphere with perfectly reflecting mirrors is a particular case of a cavity resonator
The others answers presume that the light ray (or a single photon if you prefer) travel along a plane that contains the centre of the sphere and the question set no constraint on this. Also the dimension of the sphere (radius) must be adequate to the wavelenght of the photon to trap it (total reflection).
There are certain configurations of angles where the ray will return to the same initial point, and I suspect that in any situation it will always return to the origin. Probably it is already proved.
Is there any arrangement of mirrors one can place inside the sphere such that the photon will never escape?
Yes - The number of mirrors, size, the shape and orientation are important and some constraints on these must be previously known. The solution is trivial if we can change the mirrors configuration after the ray is inside or with a proper rotation of the sphere. I am not visualizing a solution if we can not change the configuration.
A cavity resonator is a hollow
conductor blocked at both ends and
along which an electromagnetic wave
can be supported. It can be viewed as
a waveguide short-circuited at both
ends (see Microwave cavity).
physicists-trap-light-in-a-bottle and bottle microresonators
Even a perfect reflection can not avoid the Evanescent_wave
so, there are no perfect mirrors. Some of the energy will evade.
In optics and acoustics, evanescent
waves are formed when waves traveling
in a medium undergo total internal
reflection at its boundary because
they strike it at an angle greater
than the so-called critical
angle. The physical explanation
for the existence of the evanescent
wave is that the electric and magnetic
fields (or pressure gradients, in the
case of acoustical waves) cannot be
discontinuous at a boundary, as would
be the case if there were no
evanescent wave field. In quantum
mechanics, the physical explanation is
exactly analogous—the Schrödinger
wave-function representing particle
motion normal to the boundary cannot
be discontinuous at the boundary.
In optics, evanescent-wave coupling is
a process by which electromagnetic
waves are transmitted from one medium
to another by means of the evanescent,
exponentially decaying electromagnetic
dielectric microsphere resonators
The whispering gallery modes (WGMs) of
quartz microspheres are investigated
for the purpose of strong coupling between single photons and atoms in cavity quantum
electrodynamics (cavity QED). ...
In optics the treatment is done using only the wave properties of the light, and as a particle when it is absorbed by some atom/electron/...
EDIT add: The first sentence of this answer is not very interesting. In one situation the study uses 'geometric optics' because the wavelength is << sphere Radius and we must use EM pure treatment in a waveguide (the wavelength is in the order of transversal dimension of the waveguide).