One episode of Bones last season had the team not attemting to catch a murderer, rather they were trying to exonerate a girl/reunite her with her birth parents.
A Yes Prime Minister episode had Hacker and Sir Humphrey working together to defeat the French.
So which episodes of series broke with the standard formula and the result was a success?
Buffy The Vampire Slayer had lots of these
“The Zeppo” - Xander has a small, crazy adventure on his own, unnoticed by the other character who are dealing with a world-shattering apocalypse, which we never get to see
“Once More With Feeling” - the musical episode
“Conversations With Dead People” - each of four main characters in separate stories where they are basically, well… having a conversation with a dead person.
Angel broke it’s basical formula before it had even set it. The very first episode ends with Angel failing to save the girl he’d been sent visions of.
Star Trek: The Next Generation did “Lower Decks” - the story of four minor, low-ranking characters as they observe (and sometimes try to get the approval of) the major characters.
Babylon 5 did “A View From the Gallery”, which had the same kind of premise as TNG’s Lower Decks. Also “Intersections In Real Time”, in which each act was a real-time scene featuring only Sheridan and an interrogator. The episode ends not with him escaping, or defeating the interrogator in a battle of wills, but with him realising that the interrogator has been duping him the entire time and that his torture is only just beginning.
Doctor Who did “Midnight”, which is not only just set in one small room with a bunch of trapped charactes, but has all the characters turn on the Doctor one by one, and he’s unable to do the typical Doctorish thing of grandstanding and babbling his way out of it. For once, no-one buys it.
Breaking Bad did “The Fly” last season, which was basically just Walt and Jesse alone in their meth lab as they chased a stray fly, with us as the proverbial fly on the wall.
Seinfeld did the backwards episode, which started with the final scene of the story (and the closing credits) and proceeded backwards in time, closing with the start of the story followed by some further flashback scenes to several years before (exolaining the origin of the story) and further back to before the start of the series.
Mad About You did an episode that was more like something from the Twilight Zone. Time somehow unwound, reality shifted, and Paul and Jamie began living the lives they would have led had they never met each other. Anyone remember this or have any details? I only saw it once but remember loving it.
One episode of House was a day from Dr. Cuddy’s perspective. In that episode, House would appear occasionally with some stupid request and seemed like an annoying pest interrupting her important work. It was great fun to watch.
X-Files had several lighthearted–even comedic–episodes. “Jose Chung’s from Outer Space” is probably the most notable, featuring a cameo by Alex Trebek as a Man in Black. Another really funny episode that used the same kind of storytelling (multiple unreliable narrators giving different versions of the same events) was “Bad Blood” (which, BTW, I see via Wiki was written by one Vince Gilligan, creator of Breaking Bad). Mulder and Scully give accounts of the same events, but with very different details. And we learn that vampires have OCD, so one can escape them by throwing a handful of sunflower seeds on the floor.
The Doctor Who episodes where the Doctor is a background character actually end up being some of the best, like Blink and Love and Monsters (I know not a lot of people like the latter because the monster is naff, and it is, but the acting is great).
Many consider the best episode of Homicide to be one where they talk to a dying man for most of the episode, instead of investigating anything, making fun of someone in the box, or their other usual things.
(Personally, the episode didn’t do much for me.)
There was an episode of “Mary Tyler Moore” where Lou and Murray are consistently losing betting on NFL games. Ted overhears them and without knowing anything about football, picks the winners for next week and does it for the rest of the season. This eats at Murray “how can an idiot like Ted be so right” and, well I won’t tell the rest and don’t know how to set a spoiler box. But the last scene is a classic.
LAW & ORDER had one classic episode open with a criminal dying in the electric chair; we then follow the cops and prosecutors who put him there, as they spend the rest of the day finding new and different ways to put off dealing with it: two of 'em wind up at the same bar and get increasingly hammered, another sleeps with someone he just met – played by a then-unknown Jennifer Garner! – and the whole thing wraps up with the out-of-nowhere season-finale ending that got Jill Hennessy off the show by putting her character on a slab.
The Drew Carey Show did this quite frequently; so frequently it could almost be said that the “normal” episodes were the ones breaking convention.
The episode title was “Up in Smoke”. I am pretty sure it was a season finale and yes it was a great episode.
I hardly ever watched Just Shoot Me but I remember they had a parody of a Biography episode that was funny.
The Simpsons did a Behind the Music parody that was good too.
Friends did an Alternate Reality episode that was good.
Most shows that last a few seasons end up doing this sort of thing. The X-Files did a Cops episode. 30 Rock did an episode that was basically an episode of Tracy Jordan’s Wife’s reality show.
The funny thing is these things are usually so out of left field and refreshing, they work more than they don’t I think. I am trying to think of one that was a dud and coming up blank.
MAS*H did a few of these
– The episode told entirely from the POV of a wounded soldier
– The episode (co-written by the show’s medical advisor) in which a clock is shown running in the lower corner, emphasizing that they only have 25 minutes to save the patient if they want him to keep his lower limbs. What was particularly neat was that the clock didn’t stop for the commercial breaks, but picked up with the elapsed time subtracted.
– The Dreams episode.
– the episode done as if it were a “newsreel” of interviews with the MASH staff.
Well, for a dud, ST:TNG did a clip episode. Shades of Gray. No other ep of any ST series was a clip show.
M.A.S.H. did a couple of interesting ones. A news reel ep, maybe more than once. A patient perspective ep. A dead guy ep.
I type slow enough for a 3 minute window of simulposting.
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MASH had several, such as a documentary where the characters were interviewed (I’m pretty sure that episode was cited here recently for the priest’s quote about a surgeon warming his hands over the exposed entrails of a soldier), an episode from the point of view of a mute patient, and at least one bottle episode, like the one where Hawkeye and Hotlips were trapped somewhere or other.
I think most long running shows will break the formula, even if only in little ways. House has always seemed extremely formulaic to me, but, as cited, they’ll shake it up from time to time.
The all time formula champ for me was Home Improvement, and while I’m willing to bet that they violated that formula, I didn’t really watch the show very often.
Ninja’d twice, by god.
Cuddy was already mentioned, but they also did a House episode from Wilson’s POV: he occasionally glimpses what would’ve been the main story (with the usual crew doing wacky stuff to diagnose their patient of the week), but we instead spend most of the episode watching him solve medical problems as the anti-House: he cures one patient by genuinely caring about grandfatherly displays of affection, and plays organ donor to help another patient beat cancer – after having donated his own blood to help the guy beat an unrelated cancer years earlier.
Plus they did one similar to the MASH episode with Mos Def as a patient with locked-in syndrome.
You obviously didn’t read the part of the OP where he said that the shows were supposed to be good. 
There’s an episode of Law and Order: SVU, where 99% of the episode is just Benson and Stabler interrogating one guy. A really good episode, especially since it came as the show was starting to go downhill.
Then there was the ER show that was done live, twice, once for the east coast, once for the west coast. Quite difficult.
The only episode I can think of that might fit is the one where Tim OD’s on Polish food and spends all night watching the early episodes of Tool Time, when he had a beard and was competent, and Al was clean-shaven and not-so.
I loved the musical episode of Scrubs. The songs were really deft parodies, and the whole thing was very polished; you could tell they’d rehearsed the bejasus out of it.