There are two really good answers. But bear in mind we are talking about drop in the ratings (that’s how I interpret this question) not necessarily drop in quality or appreciation by critics.
The first is the 1960s version of “Batman.” When it premiered, it quickly went to #1 in the ratings and stayed there. The extremely predictable formula of a cliffhanger on one night, followed by the incredible escape on the next night, added to its immense popularity. For kids it was exciting adventure, for adults it was a humorous farce and great fun.
But the very predictability of this format got old extremely fast. Part 1 of every two-parter showed the dynamic duo put into some elaborate death trap. After a while, even little kids started wondering “Why doesn’t the Riddler just take off his mask… or maybe just shoot him?”
And the campiness of Batman was great fun but eventually the jokes grew old. Holy cliche, Batman!
For whatever reason, the formerly #1 show, by the third season had ratings in the toilet. Even the alluring presence of Yvonne Craig, as Batgirl, couldn’t save the show, despite her adding a lot. Women got a super hero to root for, and men got… er… eye candy. It still wasn’t enough.
The second case of a show having a big drop — and this is instructive for another reason — was Twin Peaks, shot practically in my backyard, east of Seattle. (Those of us who live where I do can actually drive out to most of the Twin Peaks iconic settings from the show, so we are extremely nostalgic about it.)
Anyway, the show was #1 in its first season, partly due to the way the question, “Who killed Laura Palmer [the homecoming queen]?” was on the mind of everyone in America.
People were addicted to it. But by the second season, the network kept putting more and more pressure on director/creator David Lynch to answer the question of who killed Laura Palmer.
The problem as Lynch himself warned, was that once you answer this question, driving the plot of the show, people stop having a reason for tuning in. And that’s exactly what happened. The answer was given half way through the second season, and after that, Lynch and his fellow writers tried to come up with NEW mysteries and NEW conflicts, but somehow it wasn’t enough.
(I should note, however, that their attempts involved bringing in a then new-actress, Heather Graham, to become Kyle McLachlan’s love interest. Sadly, it wasn’t enough.)
Postscript: Batman was never revised as a live action television show, but went on to be the basis for the best films in the “DC Universe”; as for Twin Peaks, it essentially got a Third Season just a few years ago in “Twin Peaks: The Return,” and I loved it. It was, once again, filmed partly in the State of Washington.