As a thought experiment, maybe time stops for a photon... maybe not. (See later in this post for my take on the answer.)
In the real world, time stoppage can't happen for persons because persons have non-zero rest mass. Persons can NEVER travel at the speed of light in an unwarped space-time region. Only an object with zero rest mass (e.g. a photon) can do so. Even a thought experiment needs to include some aspect of real-world properties - otherwise it is merely science fiction.
In order for your thought experiment to have meaning, you must include some sort of warping of space-time to allow you to violate the implied infinity of energy required to bring a real object to the speed of light - because in your E=mc^2 equation, dilation ALSO affects mass. As speed increases, time-flow decreases but effective mass INCREASES. This also begs the question of whether the speed you actually reach would be the speed of light in that warped space-time volume of space-time.
Since we are dealing with non-zero rest mass for the object doing the traveling, our thought experiment must take into account that current theory says that object can't reach the speed of light. Therefore, time would not stop, it would just dilate a lot.
Now, here is the 64$ question - does time flow for a photon? My counter-question is, how would we tell?
Well, the real difficulty in answering this question is that time doesn't exist independently. It is a measure of change, a concept of mutability. Movement is a form of change and we measure time based on movement (sometimes, as e.g. a light-year's true meaning). Vibration is a form of change (repetitive though it might be) and we use atomic clocks to measure time based on vibration. Radioactive decay is a statistical process that can be used to express time in terms of half-life periods. Chemical processes reach natural equilibria based on time and reaction probability. I.e. a reaction with a lot of energy to "spur it on" occurs quickly. Low-energy reactions are slower. That ticking object on the mantle or desk measures time as a function of the conversion rate of potential energy (in a wound-up spring) to kinetic energy (in the mechanical clock's escapement gears).
In vacuo, a photon does not appear to change. That is, photons from close stars and photons from distant stars seem to have the same properties even though our understanding of light says that the light from the closer stars is younger (by a lot) than the light from the farther stars. If there were no apparent changes in the nature of the photons, we would have to say that we have no basis for any assumption of time flow in those photons.
You know of course that I can't leave it there. This line of thinking opens up a REAL can of worms. Since this is the Watercooler, let's have a typically wild digression. Suppose that light really WAS somehow changing by interacting with time... what would such change look like?
According to our theory of light interacting with something, it can only do so by losing or gaining energy because it has no rest mass to be changed by the interaction. But if a photon's energy changes, that light's corresponding frequency has to change as well. The laws of thermodynamics suggest that in vacuo, if light were to change somehow, it could only lose energy, never gain it, because you can't create energy from nothing. Note that adjacent pairwise particle creation is symmetric - one particle and one anti-particle, so there is balance. Photons ON THE AVERAGE would not gain energy in the presence of spontaneously generated particle pairs. More precisely, whatever one photon gained, another photon could lose.
So... here's the question to be thrown into the mix. Let us suppose for this argument that light can decay over time. It means that it would lose energy and would therefore undergo a shift towards redder colors (and towards infra-red as it gets further.) So... do you want to propose to an astronomer that Ed Hubble was wrong about the meaning of the red-shift experiment?
The stars aren't moving apart at all... it is that LIGHT is decaying and in so doing is red-shifting based solely on distance traveled.