If someone traveled for one year at the speed of light, how many years would pass on earth in that time

Taking into account time dilation. Hypothetically.

You cannot do a time dilation calculation for the speed of light- as you probably know, nothing can go at the speed of light itself, and indeed, the formula for time dilation breaks down if you try to do so- if you put your speed at exactly equal to the speed of light, you get a division by zero in the formula and hence no result.

You can, however, go arbitrarily close to it, and the formula will give you a result. For instance, if you went a 99.9% the speed of light (written as 0.999c), then for every year you experience, people on Earth would experience 22.36 years.

This website here does the formula for you automatically (which is where I got the above result):

http://hyperphysics.phy-astr.gsu.edu/hba…

Your correct, they would age differently, but it wouldn't be great. At most it would be no more than 10 years or so but I don't know the math/ratio of time dilation at light speed to stationary on Earth. I do know the Tau difference wouldn't be that great at a distance of just one year. For true Time Travel, go a great distance, pick up Tau and the ratio will become extraordinary. Read Tau Zero for a story of a spaceship that can only pick-up speed, until Universes are born and die in only a day on the ship. Pretty good story.

http://en.wikipedia.org/wiki/Tau_Zero

Knowing what the word "if" means, the answer to your question is "An infinite length of time". Were you to reduce the time traveled at the speed of light by your someone to a trillionth of a second the Earth would experience and infinite length of time relative to your someone.

And before the TD's arrive, what would t's limit be as v approaches c?

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The others are wrong. If you travel for one year at the speed of light, one year of time will pass on Earth. If you travel for two years at the speed of light, two years will pass on Earth. You will be exactly one light year distant in the first case, and two light years distant in the second case. Photons travel in this manner all the time.