Although very long lived, most dynasties - from the carolingians to the qing - came to an end eventually. your job here is to delay that for as long as possible - starting in middle antiquity, have a dynasty or a powerful family survive, in power, until modernity. their power can wane, of course, but they must survive and hold on to some actual power (no puppet emperors or anything)
Who's the most likely? I like the idea of an ancient roman family like the Cornelii or the Julio-claudians, but other interesting possibilities include indian (a Mauryan survival would be fascinating) and chinese dynasties surviving until modernity.