If photons never experienced time, how do they exist? If photons are said to always travel in a vacuum at speed of light, then according to the time dilation equation, photons shouldn't "experience" time.
If they don't experience time, doesn't that mean that they shouldn't actually exist, as, without time, beginning (creation) shouldn't make sense?
If something has ever existed, shouldn't it experience at least some amount of time?
Shouldn't photons experience a very small amount of time when it was created?
 A: You are trying to apply your everyday intuition to photons. But your everyday intuition of how the world works has been developed from years of dealing with macroscopic objects such as tables, chairs, cups etc. It is therefore a very unreliable guide when it comes to predicting the behaviour of objects such as photons. So we have to rely on mathematical models (validated by experimental results) to guide us instead.
In a comoving frane of reference, a photon is indeed created and destroyed at the same instant in time. But in the frame of reference of an external observer some period of time elapses between the creation and annihilation of the photon, so there is a period during which the photon "exists" for the external observer. This is just an extreme example of the relativity of simultaneity - events which appear to be simultaneous in one frame of reference need not be simultaneous in another reference frame.
A: If you were in the photon's place you might perceive time and distance differently. If a photon is created during an interaction between particles it might travel through the vacuum in a certain way from one person's perspective. Depending on where you are relative to the photon you might observe it having a different frequency and therefore a different color. Same photon - different perspective -different color. We're putting you in the photon's place for now though. At the speed of light time-dilation (time-slowing down) is 100%. That means as long as you travel at light-speed no time passes. At light-speed length-contraction (length-shortening) is also 100%. So, at light-speed there is no distance to another object. From your perspective as a photon then all the Universe is where you are all the time. This is the same condition we say existed before the Big Bang. So, it is only us who perceive differently. For photons it is always the same : )     
A: In relativity, there is no rest frame of photon in ordinary sense, so the whole question of photons experience makes no sense. No matter what you will do, the photons frame is unreachable and you are basically left only with experience of how the photon acts in other frames. In those frames, the time is well defined and all the processes - like photons creation - are "normal". The question how photon itself experiences those processes is, in relativity, outside of the domain of physics. 
That being said, abstractly one can indeed define frame that is comoving with photon. In such frame, there will exist parameter (coordinate) which will indeed count the passage of photons life (abstractly, because as far as I know there does not exist any physical process that could do the counting in the photons frame, so any counting would be only in mathematical sense) - as the time is parameter that counts yours - and with such parameter one can describe the processes happening to the photon. However such parameter behaves differently than time as we understand it, so we cannot call it time. But from philosophical perspective, I guess it is all right, because the passage of photons life is still parametrized, even though it does not have such nice properties as time does. In particular, there are problems with synchornization of the parameter, because any signal sent from the photon to some place could never return back. So basically, you can define "time" for the photon, but the time would be photons own and it could never compare its "clocks" with anything else. 
