Light that travels eternally because of inflation?
Infinities are a type of singularity. As soon as you introduce an actual infinity into any kind of mathematical model you can't expect it to make useful predictions. Mathematical models can't deal with infinite time. Similarly, using a frame that moves at the speed of light, with infinite rapidity, is singular.  

In the particular case you describe, in a regular frame, the photon becomes progressively redshifted by the expansion of spacetime, its energy (to an observer in a inertial frame) becomes lower and lower.