Think about doing long division. We append a 0, get another digit in the quotient, multiply and subtract to get a remainder. We can assume that remainder is not zero because then we wouldn't have a repeating decimal. At each step the remainder has to be less than N, because we're dividing by N.
So there are N-1 possibilities for the remainder. On the off chance we generate N-1 digits and N-1 remainders and none repeat, we know the next one will repeat, because we've already generated all the possible remainders.
So the largest size of the repeating portion is N-1. This maximum happens first for 1/7.
