For example, 1/3 is fairly easy as 3 repeats from the first decimal point. But suppose the number is 12.345678678678678... How can one efficiently ask Mathematica to return {6,7,8} in such a case?
Just a slight modification to ilian's post, if all you wanted was the repeating digits
Last[First[RealDigits[4111111/333000]]]
Paul.