This is of course inspired by the death of Natasha Richardson, third generation of the Redgrave acting family. Granddaughter of Michael Redgrave and Rachel Kemp, daughte of Vanessa Redgrave and Tony Richardson, all of whom were in the business.
Let’s list some others.
Walter-John-Anjelica Huston
Ozzie-Ricky-Gunnar/Matthew Nelson
Jawaharlal Nehru, his daughter, Indira Gandhi, and his grandson, Rajiv Gandhi… all Prime Ministers of India
Do all three generations need to be famous or just 1st & 3rd? If the latter, there’s John/Lionel/Ethel Barrymore (1st gen) and Drew Barrymore (3rd gen). The intervening John Drew Barrymore (2nd gen) isn’t really that famous in his own right.
Senator Prescott Bush, President George H.W. Bush, President George W. Bush.
Drew Barrymore is apparently fourth generation in the movie biz - her great-grandparents Maurice and Mae Costello were both in silent films.
Henry Fonda, son Peter, and granddaughter Bridget.
John Carradine was the father of Keith Carradine, who was the father of actress Martha Plimpton.
Jackie Gleason’s son-in-law was actor/playwright Jason (“That Championship Season,” “The Exorcist”) Miller, and his grandson is actor Jason Patric.
Joseph/John/Caroline Kennedy.
Unless you consider them royalty, that is. 
Closest thing we Yanks have to it, indeed. Shouldn’t the fourth generation be making a name soon?
Are there ary third generation rockers?
MLB baseball players Ray Boone -> Bob Boone -> Aaron & Bret Boone
Debby Boone’s Father is Pat Boone. Her (maternal) grandfather was Red Foley.
Well, there’s Hank, Hank Jr., and Hank III.
Violin virtuoso Efrem Zimbalist was the father of “The FBI” star Efrem Zimbalist Jr., and the grandfather of “Remington Steele” actress Stephanie Zimbalist.
Maybelle Carter, June Carter Cash, and Carlene Carter -country music singers.
Loudon Wainwright Jr. was an editor and well-known columnist for Life.
His son, Loudon Wainwright III is a successful singer/songwriter
His son Rufus Wainwright and daughters Martha Wainwright and Lucy Wainwright Roche are also successful singer songwriters.
Also along these lines, painter Lucien Freud is the grandson of Sigmund Freud, but I’m not aware that Ernst Ludwig Freud, Sigmund’s youngest son and Lucien’s father, was particularly well known.
Also if you’re willing to skip a generation, William Henry Harrison and Benjamin Harrison.
Cornelius Vanderbilt III -> Gloria Vanderbilt -> Anderson Cooper
Sort of American Royalty…
I don’t think the father and grandfather count as famous, but I read that Joss Whedon is one of a very rare breed–a third generation television writer.