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No best answer has yet been selected by blinkyblinky. Once a best answer has been selected, it will be shown here.
For more on marking an answer as the "Best Answer", please visit our FAQ.The origin of mass is a bit of a mystery and no-one yet knows what causes it (although there are several theories, such as the "Higgs Field" see: http://en.wikipedia.org/wiki/Higgs_field). Suffice it to say that some elementary particles have mass, and some don't.
Those particles which have mass--such as electrons, protons, etc--can never be accelerated to the speed of light, because, as you point out, this would require an infinite amount of energy.
But particles that do not have mass--such as photons--have to travel at the speed of light. Massless particles are not allowed to travel at any other speed. So photon's cannot be "accelerated" but must be travelling at the speed of light as soon as they are emitted.
A photon is massless which is why it moves at the speed of light the instant it's created. It does have a momentum proportional to it's frequency without needing a mass.
Interesting that you should ponder why very old photons don't slow down. In a way they do because the universe has stretched since they were born, but rather than the momentum being scrubbed of the speed, it's scrubbed off the frequency, causing them to appear redder (red shift). If you were to ride with the photon the universe would appear to foreshorten along your line of travel and the photon would appear to take absolutely no time at all from creation to collision, even if an outside observer measured you as travelling across intergalactic space for billions of years.
You can see what happens when light does slow down if you notice how a straight stick appears to bend in water, or how diamonds split white light into colours.