The second-to-last line in the opening song is interchangeable, but almost always rhymes with "totally insane-y" (the exception being Yakko saying "uhhhhh" as if he's at a loss for words). To note, however, when the show was aired on Nickelodeon, this gag was removed and replaced with Dot saying "Nickel-eeny" (which didn't even rhyme with "aney") for every opening.
Animaniacs also includes a gag-credit amongst the regular credits of every episode, most commonly "Goodbyyyyyye, nurse!"
This becomes amusing on VHS releases as they have all of the alternative openings, one right after another.
Also, in early seasons, the line Wakko packs away the snacks was followed by While Bill Clinton plays the sax (showing an image of the then President). This changed to "We've got wisecracks by the stacks" in Season 2, and in later seasons it showed the Warners surrounded by paperwork while they sing We pay tons of income tax.
On the same subject, after Pinky and the Brain was spun off into its own show, the line "Meet Pinky and the Brain, who want to rule the universe!" was replaced with "Meet Ralph and Dr. Scratchansniff, say hi to Hello Nurse!".
Stan will pick up a paper in the opening credits, and the headline will vary every time. It will usually be a cynical pun on a common aphorism, such as "Optimist Drowns in Half-Full Tub", or "Hooker Killed for Heart of Gold". Subverted in an episode where the headline is "Alien Spotted", with a picture of Roger, thus jump-starting the plot of the episode (and ending the theme song early).
Starting with the third season there's a new opening which ends with Roger appearing in Stan's car and stealing the last line. The new Couch Gag is how Roger is dressed.
Some Fox affiliates, when re-running early episodes, skip its normal opening and instead play just the couch gag of a post-season 3 episode (in most cases the one from "1600 Candles", where Roger is not wearing any disguise).
Near the end of the opening credits, Jay will watch a short trailer for a fictional movie, which will usually be a sequel to an existing movie, or a cross-over between two movies. Regardless of the clip, Jay will declare "It Stinks!". Also, the credits open with Jay being roused from sleep by a phone call, always bad news, to which he reacts silently; this was later replaced by assorted talk radio chatter from his alarm clock.
The opening credits of Dave the Barbarian finished with the castle collapsing as the door slammed, revealing something different behind the wall each time.
The Emperor's New School opens with Kuzco (or rarely some other character) making a weird observation about the coming episode. He thens yells "Theme music!" and the opening credits start.
The Fairly OddParents ends its opening titles with Vicky's heading getting changed into something (say, a toilet). Starting around season 2, Vicky's head was changed into something that had something to do with the plot of the episode.
At the beginning of the opening credits, a small tagline is shown underneath the Futurama logo (for example: "Now in Smell-o-vision," "As Foretold by Nostradamus," or "From The Makers Of Futurama").
At the end of the opening credits, the ship crashes into a large screen that is playing a different classic cartoon each time. The majority of the episodes from after the show was Un-Canceled skip this part because the commercial time became longer.
Of course, he does have the advantage of not moving his mouth when he speaks ("Wouldn't I make a great ventriloquist? My lips never move."). (One particular airing went out with silence where Garfield's quip usually went. See it here.)
Apparently, this was a mistake by the network. It was fixed later on, though.
The opening of all Pepper Ann episodes had her find a different random object under her desk each time, from $5.00 American to the TV remote.
In the title sequence of The Proud Family, Oscar closes the door on a series of characters, always starting with Penny's friends and ending with Suga Mama, who forces the door open and goes in. Sometimes one or two recurring characters appearing in that particular episode will appear after Penny's friends, sometimes it just goes straight to Suga Mama.
Every episode of Sheep in the Big City starts with a random commercial parody before the opening sequence.
Can you find Emmitt Nervind X times in this episode?
Freakazoid also had a semi-couch gag in the opening song around the word "Freakazoo." Some episodes would show Freakazoid behind bars while this word was said; other times an ad for a free kazoo was shown instead.
The "Chimpanzee" lyric of the theme was always accompanied by a different live-action clip of a chimpanzee - except on the rare occasion when they'd show the animated chimpanzees instead.
The credits of The Venture Bros. always had a "nickname" insert between the first and last name of one of the animation supervisors that is composed of an off-hand line from the previous episode.
The opening of Chowder is from the point of view of inside an oven. While the activities in the foreground are always the same, the living food being cooked is different in each episode.
Or at least that was the intention, each episode did have a different food being cooked but Cartoon Network put the same intro on the majority of episodes so this little couch gag is mainly heard of but rarely seen.
The Secret Show opens with Agent Ray evicting Granny from in her timeslot in a different fashion every episode.
The opening credits for Invader Zim always ended with the disguised Zim imagining himself laughing maniacally atop his house, oblivious to his surroundings. There were only three scenes used interchangeably for normal episodes, but two unique ones were made for Halloween and Christmas.
The extended scene at the beginning of every episode of Jackknife (the prisoner with the pompadour haircut) getting arrested again by Jailbot and carried through a variety of bizarre events back to Superjail. This effectively make most of the opening a Couch Gag, and the only Stock Footage is the last couple seconds of going through the cloud shaped like the Warden's head to Superjail.
Further playing with this, the things that happen in the scene sometimes affect/set off the plot of the episode. One episode started with a particularly brutal serial killer who ripped off Jackknife's face which resulted in Jailbot taking him instead, and other was centered around a kid that Jailbot accidently picked up.
The title cards for every episode would have Porkchop painting the episode's title under Doug's logo, adding in "'s" to the logo if it was needed (sometimes it fades in by itself while Porkchop is painting the title), with Doug always reacting negatively enough to scare Porkchop into running offscreen, though the outcome of this would vary from episode to episode. When DisneyRetooled the series, Porkchop's way of writing the title would correspond with the episode's plot.
Additionally, the way that the screen became black after the title was shown varied as well: sometimes Porkchop turned off the lights, sometimes Doug did, and sometimes Porkchop would throw the rest of his paint onto the screen.
On The Wild Thornberrys, the opening sequence ends by zooming in on a map of the world, to the location where the Thornberrys were in the current episode.
Starting in season 2 of Blues Clues, Blue hides in a different spot outside the house.
The mildly obscure Disney series The Buzz on Maggie has the title character dress up in two different outfits/costumes in each credit sequence.
The short-lived Clerks: The Animated Series had two couch gags. Before the opening credits, a disclaimer (inserted via Executive Meddling) would run, followed by the announcer making an off-hand comment like, "Is anyone still watching after all that?" After the opening Theme Song, Randall would make a sitcom-like announcement about a supposed "live studio audience" as if it were a live-action show. (i.e. Clerks is drawn by a live studio audience.)
For Robot Chicken, the Stoopid Monkey Vanity Plate is different each episode; all of them are a still picture of a monkey being Too Dumb to Live or already dead. For the first half of Season 5, episodes only featured the standard Stoopid Monkey logo that was introduced in Titan Maximum. The second half of the season featured Fan Art style pictures.
On the first season of The Brak Show, the beginning of each episode showed the title of an old television show, with one word changed to the name 'Brak' (i.e. Leave it to Brak).
The intro to Buzz Lightyear of Star Command would sometimes have a humorous incident occur when Buzz gets to the TV (i.e. he smacks right into the TV, or Wheezy and RC knock him away). Fittingly enough, the montage of clips from the episodes would vary depending on which gag they used.
The Super Hero Squad Show's opening features a scene where The Incredible Hulk grabs the Infinity Sword and transforms into a different version of himself in every opening, from a baby version to Joe Fixit, to the Grey Hulk.
For the first few seasons, the opening theme would include clips from that very episode, interspersed with clips of the founding members. Sadly, they dropped this for later seasons.
The title of Sealab 2021 would appear in the opening credits as "Sealab 2020", the name of the show on which it was based. The changeover of the final "0" to a "1" would be accompanied by a different sound effect each episode.
Perfect Hair Forever took this trope to the extreme with both its begining and ending credits having exclusive sequences with each episode.
In a similar manner to Garfield and Friends, the televised version of Veggie Tales's opening theme ends with Pa Grape making a different comment about Archibald's sweater.
The episode title cards from Arthur usually show a different animation involving characters from the show interacting with a large circle. For example, one title card shows Francine hitting the circle as if it was a gong, followed by Arthur screaming in annoyance. Another title card had Buster spoof the MGM logo.
Family Guy has a few episodes with an alternate opening, all made right around the same time. Presumably, the idea was to make the entire opening a couch gag, but this never really materialized.
Teen Titans has a running title gag that, whenever they did a silly episode, a nonsensical version of the theme song would be sung in Japanese.
Hard to catch, but in the opening to Bob's Burgers, the store to the right of the title restaurant changes from episode to episode. These tend to have punny names (e.g. "I'd Hit That Boxing Gym".) Also, the exterminator's truck that comes when the restaurant is infected by rats has a different name and logo.
In the DVD version of Cars Toons, Mater will actually say "If I'm lyin', I'm cryin'" in a different manner that relates to the episode's plot. For example, in "Moon Mater", it is nighttime, and Mater says "Houston, if I'm lyin', I'm cryin'", and in "Mater: Private Eye", it is black and white, and Mater will say the opening title in a manner similar to film noirs.
T.U.F.F. Puppy has several different openings with some random thing happening after Dudley strums the bass chord at the end. Either he falls off of the logo, the bass explodes, the bass turns into a rocket that carries him offscreen, or the bass turns out to be The Chameleon who Dudley then chases offscreen. There's also one where Kitty plays the bass and Dudley drives into her on a motorcycle knocking her offscreen, followed by her coming back onscreen giving him an angry look, one where Keswick plays it and it turns into a robot and chases him away, and one where Snaptrap plays it and Dudley drops down on him and takes it, after which it explodes again.
Tripping the Rift: At the end of each opening sequence, Spaceship Bob (the voice of the computer) would utter some different, humorous sentence.
Near the end of the opening credits for Mr. Bogus, after the part with Bogus and the ice cream has played out, it will then segue into a different claymation vignette, and play a second one after the first one has aired. After the second vignette is finished, it then cuts to Bogus who is doing his trademark dance before the show's logo comes up at the end of the intro.
The opening of Good Vibes has Mondo and Woody hallucinating something new every time they run through Lonnie's smoke.
The opening of Moral Orel has Orel waving to God (towards the sky) in different ways for most episodes. The credits feature Orel making a stop-motion animation film, although this happens less as the show goes on. In the Wham Episode, Orel does not look up towards God, but instead remains in prayer.
The first season of KaBlam! had a segment called "Surprising Shorts", where it would open by showing June pull down Henry's pants revealing that he's wearing embarrassing boxer shorts printed with anything embarrassing from bumble bees and puppy dogs to Bob Dole!
The Addams Family cartoon of 1991 had Wednesday trying to pull a prank on Pugsley, if she succeeded or not and how was different every episode. Also the photo shot of the family at the end is different.
Each theme song of Peg + Cat has Cat doing something related to the episode's topic.
Newer episodes of Squidbillies all open with a different cover of one of the two songs played during the intro in the early seasons.
The opening of The New Mr. Peabody & Sherman Show features a shot of a museum, the banners on the outside change every episode to advertise whatever historical figure the focus of that episode's time travel story is.