A photon cannot be said to have its own inertial reference frame, because inertial reference are defined to be a family of coordinate systems that satisfy the two fundamental postulates of SR, one of which is that light moves at c in all frames. You could construct a coordinate system where the photon was at rest, but since this coordinate system wouldn't be an inertial frame there'd be no reason to expect equations that are specific to inertial frames, like the time dilation equation, would still give correct predictions in this frame.
One thing you might try is considering the limit as an inertial object's velocity approaches c relative to the rest frame of the galaxy, and it's true that in this limit, clocks with low velocities relative to the galaxy would approach being completely stopped. Another point is that if I'm moving at some very large fraction of c relative to the galaxy, not only do I measure the galaxy to be highly compressed in the direction I'm going, but I also see clocks on either end of the galaxy as wildly out-of-sync due to the relativity of simultaneity...if the galaxy is 100,000 light years long in its own frame, and I'm moving at speed v relative to it, then two clocks at either end of the galaxy which are synchronized in the galaxy's frame will be out of sync by (100,000 ly)*(v)/c^2 in my frame. So, in the limit as v approaches c, clocks on either end of the galaxy are out-of-sync by 100,000 years, so at the same moment that it's 2014 A.D. on the leading edge, it's 102,014 A.D. on the trailing edge. And yet in the limit as v approaches c, the distance between clocks along the direction of motion is compressed to zero. So in the limit, perhaps you could say that the photon's entire history is traversed instantly, since it's going zero distance and all the different clock-readings it passes are squashed together on this zero-length path.
However, I think there would be aspects of this limit that wouldn't really be well-defined, in that they would depend on the details of what are the series of cases that you are using to construct the limit. For example, consider a particle A moving at some v < c, and a photon B moving at c. In the limit as v approaches c, both A and B are moving at c, and B is still moving at c in the frame of A. On the other hand, consider two particles moving at the same speed v, at rest relative to one another. In the limit as v approaches c, both A and B are moving at c, and B is at rest in the frame of A. So, the question of what velocity A would measure B to have "in the limit" would depend on which type of limit you used, even though in both cases they both have a velocity of c in the limit.