I'll name you three more, from the other side of the world: in chronological order, the Fujiwaras in Heian Japan, the Qings (Jurchen Manchus) in China, and the Nehrus in India. Moreover, even today, there are essentially two families in Japan who have been flip-flopping the Prime Minister position since the mid-1800s.
But your claim about the Japanese prime ministership is news to me, and a surprise given the major overhaul Japanese politics received from the shogun Douglas MacArthur in the 1940's, howbeit his influence was deliberately kept brief.
The Nehru-Gandhi family is a good get; most people wouldn't think of that, and I didn't offhand. Another is the Perons and Kirchners in Argentina, and of course the families of various other caudillos in South and Central America like the Somozas.