But Anna Freud, Sigmund’s daughter, was very famous. So the second generation was quite well represented.
Although Charles Francis Adams doesn’t have the same name-recognition these days as grandfather John and father John Q., both U.S. presidents, he was well-known in his own time, serving as a member of Congress and later as ambassador to Great Britain. He is considered to have played a major part in keeping Britain neutral during the Civil War.
Another one: Lionel and Alfred Newman, brothers, were both Oscar-winning film composers (among other accomplishments), as is their nephew Randy. Lionel’s grandson Joey hasn’t reached that level, but he’s only 33 and has worked on several film, TV, and video game scores.
US President William Howard Taft was the father of Senator Robert Taft of Ohio, who was the father of Senator Robert Taft, Jr. of Ohio and his son was governor of Ohio.
Kirk Douglas
sons Michael and Eric (now deceased) = actors
sons Joel and Peter = Producers
Michael’s son Cameron is also an actor though not as well known as his father and grandfather.
And Charles’ son Henry Adams wrote “The Education of Henry Adams” which won a Pulitzer Prize and was named best non-fiction work of the 20th century by Modern Library.
Trivia: According to IMDB, Annabeth Gish is the granddaughter of Lillian Gish, but not that Lillian Gish.
John Carradine, his sons David and Keith, and Keith’s daughter Martha Plimpton, all successful actors.
If we’re talking Freuds… Lucian’s daughters include novelists Esther Freud and Susie Boyt, fashion designer Bella Freud and Jane MacAdam Freud a painter. His brother Clement is well known in Britiain as a broadcaster writer and politician and two of his children are broadcasters and writers. Lucian and Clement’s father Ernst was an architect of some repute.
Whedon’s dad wrote for Sesame Street, didn’t he?
I am not a Whedon fan, but I’m a huge Sesame Street fan 
Don’t forget Tracy Nelson, actress (Father Dowling mysteries. She’s Ricky’s daughter and the sister of Matthew and Gunnar.
You think I would have thought of them, being related and all on the Howard side–his mother and my great-great-great grandfather were siblings.
In addition to David and Keith, another of John’s sons is Robert Carradine, best known as Lewis in “Revenge of the Nerds.” Robert’s daughter is Ever Carradine, a successful TV actress – most recently appearing in the current season of “24.”
Those Carradines are everywhere.
Perhaps stretching the definition of “famous” for the purposes of this thread, four generations of Fords have run the Ford Motor Company: Henry Ford > Edsel Ford > Henry Ford II > William Ford, Jr.
I’ll be damned. I’m sure I heard celebrity coverage of her mention that she was the granddaughter of the famous one.
Lord Randolph Churchill was a prominent British statesman - an aristocrat, but not royalty. His son Winston followed him in Parliament and gained modest fame during WW2. 
His grandson Randolph and his great-grandson Winston were also both MPs, and there are still Churchills in the news today. At least four or five generations of fame, depending on how you count 'em.
If you are into NASCAR racing:
Lee Petty, Richard Petty, Kyle Petty and Kyle's son Adam would have made four
Any others in sports?
And you can throw in Edward Bernays.
It’s somewhat sprawling than the Freuds, but the Darwin-Wedgewood family tree is worth noting. Robert Darwin is arguably not famous in his own right, but his father was Erasmus Darwin and his wife was the daughter of Josiah Wedgewood. In the next generation, he had Charles as a son and Francis Galton as a nephew. In the generation after that, Charles’s sons George and Horace became eminent figures in their respective fields. In the next, Ralph Vaughan Williams was a great-grandson of Robert. And in the generation after that, there’s the poet and Spanish Civil War casualty John Cornford.
Furthermore, the family also married into the Huxley - no slouches themselves in this sort of game - and Keynes families. And the Benns, who have produced three consecutive generations of Cabinet members.
Tex Ritter > John Ritter > Jason Ritter
StG
French impressionist Pierre-Auguste Renoir; his son, film director Jean Renoir. But then Jean’s son was an obscure English professor at Berkeley, so it kinda petered out.