If you refer to the fag end of something, you mean the last part of it, especially when you consider this part boring or unimportant. He never had much confidence in his judgement at the fag-end of the working day. the end of a cigarette that someone has finished smoking (fag (15-18 centuries))
the frayed end of a length of cloth or rope the time of the last part of something; "the fag end of this crisis-ridden century"; "the tail of the storm