First, be careful with your Lorentz factor (γ). You are missing a sign on an exponent there, which is pretty critical in this case. We usually write

γ = ( 1 - ( v / c )^2 )^**-**1/2

and so γ diverges in the limit where v->c, and then you can see that the Lorentz transformations are not defined at v=c. But we shouldn't get too hung up on this, as this doesn't really address your real question:

>I would suggest that that's a limitation of the formulation, not necessarily a reflection of reality.

And to do that, we should step back from the math for a second and think carefully about applicability. Even if we have a quantitative description of a phenomena that gives a real, non-divergent answer we must be very careful that it is actually applicable to a given situation so as not to over-extend a model. Not all answers given by an equation are correct: sometimes we're just doing math and not physics.

In this case, we build a Lorentz transform by comparing two valid inertial reference frames. One of the postulates we use to construct one such frame is that the speed of light is invariant for all observers in any frame, which leads to the Lorentz transformations. However, if we try to construct such a frame at v=c we encounter a paradox: light moving parallel to this frame must be moving at c and also must be stationary in the frame. This cannot be, so we cannot construct the frame and without the frame the Lorentz transformations are meaningless (and also undefined as the Lorentz factor is undefined at v=c).

As such, in this case, it is quite the opposite: that the Lorentz factor is undefined at c is not an artifact of the mathematics, but a reflection of something fundamental to relativity.