The great Wikipedia will help with this:
Jumping the shark is a colloquialism used by TV critics and fans to denote the point in a television program’s history where the plot spins off into absurd story lines or unlikely characterizations. These changes are sometimes the result of efforts to revive interest in a show whose viewership has begun to decline. In other cases, the changes have other causes (e.g. an actor quitting, different writers being hired) and the diminished interest is the result. The two causes can feed into each other, leading to even greater changes and even lower popularity
Now with that established, I’d like to point out the TV shows I do watch that have or almost did this…
The Simpsons. Precise time: end of Season 9. Look I love the Simpsons. The Simpsons first eight seasons are some of the best work you will find anywhere (that’s right). But these recent episodes are crap, pure crap. I can’t even watch the Simpsons anymore. Yes, the movie was okay but we are at Season 21 and it’s just not working anymore. I’m being generous about Season 9 too. Most of the time if you look at any Simpsons episode from Season 8 or earlier, you’re likely to find a real classic that grows with being older to catch even more references you might not have understood when you were younger.
The Simpsons legacy is almost with its shows but also with its crew. Many people have gone on to do great things after working on The Simpsons and they should be proud of that, but it is not working anymore.
The Office. Precise time: Season 5 (almost). The downfall of the entire world economy probably is the biggest thing that helped The Office save itself from the most outlandish subplots for an entire half season. There are some things that are nice in this first half of the season but the plot lines got so out of hand, it bordered on ridiculous. The fact that this show probably saved itself by inserting a new character to drive out Michael, and then push him right back in, might have actually saved the show.
South Park. Precise time: You Never Know. Given the world’s craziest production time (6 days per episode, written/edited/voiced-over/rendered/re-edited the week of), South Park always has the chance to rebound from a streak of weird episodes to good ones. I don’t think South Park is over the hill just yet but it is getting challenging to deal with poking fun of every single reality TV show. Look reality TV is the fall of man, I agree, but they already make fun of those too much.
However one of my favorites is “Cancelled” solely because it deals with the idea of this:
(alien character) Yeah, a show should never go past a hundred episodes, or else it starts to get stale with ridiculously stupid plotlines and settings.