I would say it probably jumped the shark when the crossovers with Petticoat Junction happened, or round about that time. There was always an element of silliness to the show but in the 7th season, Fairchild, Ellie's bear, was given almost human qualities other than being able to talk. The bear drove Mr. Drysdale to Hooterville in the old truck for instance. It just became rather outlandish.
The 8th and 9th seasons really took a weird turn. In the 8th season, the Clampetts went to visit "home" in Silver Dollar City and several new characters appeared that had been (supposedly) life long friends of the Clampetts but had never been mentioned prior with the exception of Elverna Bradshaw, who was played by Elvia Allman and had previously appeared in the 1st season. Elverna and Shorty Kellums wound up living in Beverly Hills at the mansion for a short time along with Shad Heller, another "back home" friend.
The 9th season got totally weird with the introduction of topics like Women's Lib and the appearance of Japanese characters who moved in with Jed and Jethro. At that point, both Max Baer, Jr and Donna Douglas were showing signs of getting older, so I'm sure if CBS hadn't have axed it during their "rural purge," it wouldn't have been too long before it happened.
CBS still seems ashamed of their rural heritage other than Andy and Gomer. Maybe Shout will pick up where they've left off.
Yes, I had forgotten just how bad "The Beverly Hillbillies" got in some of those later seasons. Thanks for reminding me about Fairchild!
I'd actually like the first two color seasons of "Hillbillies" and then I'd be done. For me, Jed inheriting castles in England was jumping the shark. The basic premise of the show is one that doesn't play well after five years -- the fact that the Hillbillies are "fish out of water" by living in the city. After five or six years of living in an urban area like Los Angeles, you can no longer look at city life with naive, wide-eyed wonderment. (Well, actually, maybe you can in LA!)