Darryl's Sequence

Walking back to the Middle School, Darryl was mentally calculating a sequence of numbers, and he asked me to guess the rule. He told me that the sequence began: 1, 2, 6, 4, 2, 0, 4, 3. I guessed that it might be the last digit of something, but I couldn't come up with the answer. So he told me that he was thinking of the nth digit of n factorial.

Now some values of n! have fewer than n digits, so in that case he would wrap around and keep counting. For instance 4! = 24, so for the fourth digit he counts: 2, 4, 2, 4, and gets a four. Or for 5! = 120, he counts: 1, 2, 0, 1, 2, and gets a two. Eventually the factorials get big enough so we don't have to wrap any more.

I noticed that this sequence of numbers will, at some point, have a very long string of nothing but zeros. The star question is: at what value of n does that string begin, and when does it end?