One of the most popular questions from our "ask a physicist" feature was, "What happens if you're driving at the speed of light and you turn on your headlights?" The simple answer: You can't. So quit trying.
At least ten of you asked some variant of this question, but Turael asked it first and arguably most succinctly. So Turael, feel free to email me so we can send you a free copy of A User's Guide to the Universe.
Sadly, physics teaches that we are forever confined to sub-light speeds.
There were several comments, in response to the original call for questions, that took umbrage at this certainty on the part of physicists. Are we just being closed-minded? There are always people who are skeptical of equations, regardless of other evidence. For example, long after the advent of nuclear weapons and power have vindicated Einstein, I still get detailed manuscripts from people every few weeks claiming that E=mc2 is wrong.
So no lengthy derivations, but I will give you a few observations, and hopefully that will be enough. Special relativity predicts that if you take a massive particle and keep applying forces on it, it goes faster and faster, slowly approaching the speed of light, but never quite reaching it. Right now, for example, the Large Hadron Collider has protons flying around it at a whopping 3.5 TeV. This means that the protons are traveling 99.999994% the speed of light, and when the LHC gets up to full power (at about twice the energy), the protons will go even faster, but even then, less than the speed of light. At these speeds, the difference between "at" and "a tiny bit below" the speed of light may seem academic, but it makes a world of difference.
We always need to accept the possibility that we could be wrong, but in this case, there's just so much evidence that we're right! The particle colliders wouldn't actually work if relativity were wrong. For that matter, neither would GPS devices. Michelson and Morley found in 1887 that light travels at the same speed to all moving observers, a result that doesn't make sense unless special relativity is correct. All of modern physics (and technology) is built on an edifice of special relativity, and so far, it's proven ridiculously accurate. In other words, you've got a very big barrier to overcome if you want to prove Einstein wrong.
Part of the reason that people are so confused about this aspect of relativity is that it flies in the face of everyday experience. If I'm in a boxcar moving 60 mph and throw a 90 mph fastball, someone standing by the side of the tracks will see the ball moving at 150 mph. It seems like the same logic should work with light. Except that it doesn't.
Strange things happen when you get close to the speed of light, and they become stranger still when you realize that your high school physics teachers (perhaps inadvertently) lied to you. Lots of you are sci-fi nerds, so I'm guessing that at least once in your lives you learned Newton's force equation, F=ma. Deciphering the symbols, it means that if you apply a constant force to a particle, it should experience a constant acceleration. Taken to its natural extremes, if I apply a force for long enough and the particle keeps accelerating, eventually it should exceed the speed of light. Voila! Newton's force equation (at least in the form it's normally written) is wrong.
But then what happens when you get close to the speed of light and turn on your headlights? From your perspective: nothing, or at least nothing special. If you held a mirror in front of you, you'd look exactly as you always have. In fact, one of the surprising things about special relativity is that if you weren't looking at all of the scenery passing you, you couldn't tell that you are moving at all.
But from the perspective of people standing on the sidelines, things look really cool. Stationary observers would notice that your entire ship (or racecar, or whatever you're driving at 99% the speed of light) is compressed along your direction of motion. If you're standing the right way, it'll look like you've lost weight and that your body has been flattened under a giant stone.
They'd also see your clocks – and your heartbeat, your speech, your computer cycles – running slow. This is true, but completely unobservable in everyday life. Typically on earth, it's an effect of about 1 part in a quadrillion, but at 99% the speed of light, you'll appear to be running at only 1/7th speed. The length contraction and time dilation conspire (out of mathematical necessity) to make your high beams move at the speed of light to somebody watching you from the side. But just as a baseball gets a boost of energy when you throw it on a train (which you shouldn't do, incidentally) the light gets a boost of energy as well. The difference is that it doesn't go faster; it just looks bluer. In this case, your headlights would be boosted into the ultraviolet.
Stranger still is the case of two spaceships traveling toward one another, each a 99% the speed of light. Common sense would dictate that the captain of each ship should see the other hurtling toward him at faster than light. Not so! One of the results of the constant speed of light is that all relative speeds are going to be less than you think. In this case, for example, each captain would see the other coming at him at only 99.995% the speed of light.
Back to the original question (which, incidentally, is so startlingly good that it's one of the ones that Einstein himself asked as a young man), what would happen if you could get up to the speed of light? As you get closer and closer to the speed of light, time gets slower and slower compared to stationary observers. So if you really need an answer to the original question, this means that if you actually hit the speed of light for real, time would stop entirely, which means that nothing could happen. But that's okay, because you can't get up to the first place.
Dave Goldberg is the author, with Jeff Blomquist, of "A User's Guide to the Universe: Surviving the Perils of Black Holes, Time Paradoxes, and Quantum Uncertainty." (Wiley: 2010). He is an associate professor of Physics at Drexel University. Most recently, he answered, "If the Universe is expanding, what is it expanding into?" on io9.