iSteve commeter jimmyriddle came up with a nifty set of minimal handshake links from Napoleon Bonaparte, who lived from 1769-1821, to a living celebrity:

On Elba in 1814, the exiled Emperor had a 90 minute meeting with the future British prime minister John Russell (1792-1878). In the mid-1870s, the ex-Prime Minister helped raise his toddler grandson Bertrand Russell (1872-1970). In the mid-1960s, Bertrand Russell convinced Paul McCartney (1942 – ) to oppose the Vietnam War at Russell’s house in Chelsea in London.

Bertrand Russell would be my guess for the man who shook hands with more interesting people across a longer expanse of time than anybody else in history. Even as a nonagenarian, he was a major figure in the celebrity culture of Swinging Sixties London, when it was the coolest city in the world. But 50 years before meeting McCartney, Russell met Ramanujan and Wittgenstein.