Sorry! This article has two quotes on the main page instead of just one.
"This sentence wishes to apologize for all the needless apologies found in this story (this one included), which, although placed here ostensibly for the benefit of the more vexed readers, merely delay in a maddeningly recursive way the continuation of the by-now nearly forgotten story line."
"The thing with an apology is that it's something precious, like diamonds. If it happens once in a while, it's meaningful. If it only rarely happens, it can be a surprise. But if it happens too much, it loses its value, and thus loses its impact."
Ritsu Sohma from Fruits Basket is a male crossdresser who basically has fits of BERSERK APOLOGIZING whenever anything whatsoever goes the least bit wrong anywhere in his immediate vicinity, or if he even picked up the slightest implication that something was wrong. Apparently, he become this way from listening to his parents apologize for him throughout his childhood.
Magic Users Club: Sae Sawanaguchi does this whenever her magic screws up. Meaning when she gets stuck in a room, and teleports things in, rather than teleporting themselves out (crowding the room further), she apologizes for almost two straight minutes.
Ai Kaga from Sayonara, Zetsubou-Sensei was specifically designed to lampshade and parody this trope. Even her name is a giveaway: "Kaga Ai" sounds like "kagai", "perpetrator" in Japanese, and she, indeed, believes that she is to blame for everything bad. Her spotlight episode in the anime cranks this Up to Eleven, where the main cast wanders into a city where nobody can stop apologizing for even the slightest of slights.
Shinji is almost the patron saint. Towards the end of the series, his psychological descent is seen in how he stops apologizing for every little thing and merely wallows in the disparaging opinion that he can't help anyone.
Meanwhile, in Hataraki Man, created by Hideaki Anno's wife Moyoco, Hiroko keeps telling her boyfriend Shinji to stop apologizing so much, fueling fan speculation that Anno really did marry an Asuka in his own life.
Canada from Axis Powers Hetalia has this Verbal Tic. To paraphrase the Ferguson brothers, Canadians have 12 ways to say "I'm sorry", but rarely apologize.
Sumeragi Subaru from Tokyo Babylon does this a lot. In Book 4 he apologizes so much that his sister tells him to stop saying sorry, and he apologizes for apologizing.
Jacuzzi Splot in Baccano! does a lot of apologizing for very little provocation, including claiming that he should be the one to apologize for being run into and apologizing to someone because he called her a good person without actually knowing if she was or not, and finally, apologizing for introducing himself
Minawa from Mahoromatic, usually because she's just dojikkoed the place up. "Gomen nasai" is practically her catch phrase.
Sena from Eyeshield 21 can go from improbably badass running back to compulsive, apologizing bowing fool in an instant if he gets flustered.
Pagaya of One Piece apologizes nearly every time he speaks, to the point where this is almost a Verbal Tic.
Higurashi no Naku Koro ni has a rather dark example: Hanyuu tends to say "I'm sorry" over and over, but because she can't be seen by anyone except Rika, the only characters who can hear are those who are becoming increasingly paranoid. It never ends well.
Wedding Peach anime: Hinagiku always must scold Takurou for apologizing all the time. (Not seen in the manga.)
In Code Geass, C.C. does this when she loses her memories.
Keroro Gunso: Rabbi is constantly apologizing for her brother...and sometimes herself. Possibly lampshaded, when she is apologizing in the third movie for not being more useful, despite being mind controlled.
Bakura from Yu-Gi-Oh! sometimes did this, at least in the dub. When he accidentally set off a Booby Trap in one scene, he started chanting, "I'm sorry I'm sorry I'm sorry I'm sorry" etc. Maybe not the wisest thing to do while attempting an Indy Escape. Possibly related to the fact that in the dub, he has an I Am Very British accent, and as the Real Life section attests, the British are notorious for excessive apologizing.
Azmaria from Chrono Crusade (who is also a Cute Clumsy Girl) does this quite often. In one scene in the anime, she tries to cook a meal and has several clumsy accidents happen as she does, and follows up every accident with kneeling on the ground and folding her hands together while squeaking out "I'm sorry! I'm sorry! I'm sorry!"
Sakurai Ryou from Kuroko no Basuke. He even apologizes to his opponents during basketball matches.
Aries of Fairy Tail has almost no lines at all that don't include an apology.
In one episode of Maburaho, Kuriko tells Kazuki that he does this too much. His response is to apologize for it.
Nichijou: Sakurai-sensei, being a cute but shy young teacher, apologizes a lot when she speaks to people. Also, a minor recurring character is a young shop clerk, who is just as shy, just as cute, and just as prone to apologizing as she is. The two of them even meet at one point, and as you can guess, awkwardness ensues, with the two of them being too shy and busy apologizing to continue a normal conversation.
The Star Trek fanfic Written in the Stars (a genderswap fic in which the Kirks, both Prime and Alt Reality, are woman) has the Alt Kirk apologizing pretty much any time she screws up. These apologies are almost always for Spock, who in turn doesn't know why she's even apologizing to him in the first place. It's a Kirk thing, as Kirk Prime later apologizes to Spock Prime.
In Flash Fog, Fluttershy spends thirty minutes apologizing to the mayor - then apologizes for apologizing so much.
Nina, the lead of Black Swan, shows this trope in spades.
Frank Tupelo's first line in the movie The Tourist is "I'm sorry." The woman who just sat down in front of him actually asks what he is sorry for.
Peter Parker in The Amazing Spider-Man beat up an entire train car full of people by accident, apologizing the entire time. (He had provoked them - again, accidentally - and when they tried to attack him, his new super-reflexes made him counter before he knew what was happening).
In The Party, Peter Sellers plays the unfailingly polite Hrundi Bakshi. After one of many gaffes he's committed, he apologizes profusely and incessantly to the host, who would just as soon ignore him.
Sorry for mentioning Literature at all
In America: The Book, Samantha Bee apologizes a lot in her "Would you mind if I told you how we do it in Canada?" sections. In a footnote, she says "Sorry for the footnote". This is a parody of Canada's reputation for excessive politeness.
In The Idiot, Lukyan Timofeevich Lebedev doesn't exactly say "I'm sorry", but will constantly criticize himself and agree vociferously with any insults flung in his direction, which has about the same effect.
In The Book of the Dun Cow, Senex the rooster apologizes constantly when he makes mistakes, which he feels guilty about since he is technically his land's ruler. This leads to a Deal with the Devil which eventually turns fatal.
Many of the characters from the Mortal Instruments series are prone to making profuse and dramatic apologies, Clary Fray most of all.
Vincent Demabrien from Doctrine of Labyrinths. Mehitabel has to tell him at least twice to quit it. A justified example, given the fact that has spent much of his life being brutalized by pimps and clients.
In Laughing Winds Annika frequently apologises, its even pointed out to her and questioned by one of the other characters.
I apologize for using examples from Live Action Television
In Star Trek: Deep Space Nine, Tobin Dax was one of these, as we see during Jadzia's zhian'tara ritual, where his personality is placed into O'Brien's body temporarily. Complete with being told to stop, and apologising for apologising.
Dean Winchester from Supernatural. Though despite being the absolute king of self-blame, he still doesn't apologize as much as Sam, who thinks nothing of stopping in the middle of a fight to say sorry for bumping into his brother (which, of course, leads to him getting choked).
As Time Goes By. Sandy dates a guy who is always apologising, sometimes preemptively just in case something goes wrong. She leaves him when she realises this is an aspect of a fetish he has.
Robin on How I Met Your Mother visits a Canadian bar in New York and tests its "Canadian-ness" by deliberately bumping into somebody to see if he apologizes for it. He does. And he offers her a donut. Marshall tests it by turning off the lights (because Canadians are afraid of the dark).
Pretty much every character in Smallville (at least in the first few seasons), but especially Clark. Most episodes finish with an apology duel, where they try to out-apologise each other. During the middle and later seasons, it seems that half the episodes have a scene in which Chloe and Clark mutually apologize and hug, e.g. Splinter, Hydro.
As the title suggests, the lead character of the Brit ComSorry!
A Royal Canadian Air Farce episode had then-Justice Minister Allan Rock apologizing for claims his government made about former Prime Minister Brian Mulroney. He then kept appearing throughout the episode to apologize for such things as broken election promises, fatty foods, long line-ups at bank machines, and all the cold weather they'd been having that winter. Finally, he apologized for making so many apologies and told the viewers that this would be the last governmental apology of the evening . . . "and for that, we are sorry."
Quite a few of the characters from Gossip Girl are guilty of this, Serena being the worst. Chuck averted it for the first couple of seasons by hardly ever apologising for all the jerkass things he did, but come season three he started apologising as much - if not more - than everyone else.
In Crownies Richard is constantly apologising "profusely" to Judge Walker, usually because he's brought in uncooperative witnesses. She lampshades it in one episode, while she's forcing him to copy by hand a list of prior convictions that he'd given her without a chance to properly format them, while dealing with a sprained wrist. He then informs her that his wrist was sprained in a bike accident on the way to her lecture two nights ago. She reveals that the ambulance that picked him up also dented her Prius. He apologises. Profusely.
Elizabeth on the ninth season of Hell's Kitchen. Television Without Pity's tagline for the episode in which she was eliminated read as follows: "Elizabeth's gone. She's sorry, you guys. She's so sorry. So sorry. Sorry sorry sorry! (Sorry.)"
A running gag on Monty Python's Flying Circus is a stop anytime to apologize for either interrupting a sketch or the poor quality of it. The album Another Monty Python Record starts off with it:
Michael Palin: Good evening. We would like to apologize most sincerely to those of you who have bought this record under the impression that it was in any way connected with the television programme "Monty Python's Flying Circus." This was due to an error in the printing stage of the album cover. This album is in fact called "Pleasures Of The Dance," a selection of Norwegian carpenter songs compiled by Oskar Tritt.
John Cleese: Good evening. We apologize for the previous apology. That apology was unnecessary and appeared on the record owing to an administrative error. This album is not, as stated in the previous apology, "Pleasures Of The Dance," a selection of Norwegian carpenter songs, but a new album from the humorous television comedy show, "Monty Python's Flying Circus."
Cleese described on Letterman that, owing to their terror that they might insult someone, Brits were so polite that if he (Cleese) wanted a piece of Letterman's toast, he would begin by apologizing to him: "Sorry, I wonder if I might be so bold as to possibly inquire if you could see your way clear to being so kind as to possibly if it's no bother giving me a piece of toast."
In The Inbetweeners, this is a source of mockery for Simon, who has to apologise profusely to two angry Londoners after Jay's drive-by shouting of "Bus wankers!" doesn't go down very well.
In the beginning of The Vampire Diaries, Stefan seems to be saying sorry to Elena for every other thing
During The Big Bang Theory, when Leonard and Penny are rekindling their relationship, they decide to list some problems, and then try to calmly deal with them, rather than getting angry. One of Penny's issues with Leonard is this trope.
Our apologies for having examples from Music
R.E.M.'s "So. Central Rain (I'm Sorry)", whose chorus is entirely made up of Michael Stipe yelling "I'm sorry!", while "The Apologist" is the same deal but with added "So sorry!"'s.
They played the two songs back to back in concert on at least one occasion "to prove how sorry we are".
In Eternal Sonata, Polka apologizes most of the times she's healed in battle, or after her turn of attacks. (She actually has two different versions of "I'm sorry" that she uses when healed.) She tends to do it a lot outside of battle too.
FireEmblem: Radiant Dawn has a conversation where Rafiel would over-apologize. Ike suggests that he doesn't need to be so soft, prompting a confused Rafiel to apologize again (this is how he was brought up in the heron tribe).
According to Nailah, all herons were like this, Reyson is an exception because he has spent to much time picking up manners from Tibarn.
Wild ARMs 4 - Yulie Ahtreide. Dear god, Yulie Ahtreide. How often she apologizes is even a point of discussion among the main party at one point! (She stops doing it eventually, though, after taking several levels in self-confidence...)
Oichi from Sengoku Basara. Oh sweet heavens for the love of GOD... Oichi!!
Oichi: I'm sorry... This is all Ichi's fault... I'm so sorry, please forgive Ichi...!
In the second game, Sophitia Alexandra is the Apologetic Attacker. This doesn't make her any less vicious, however.
The entire Alexandra family in the Soul Series can be called this- in the fourth game, Cassandra (Sophitia's sister) is the Apologetic Attacker though, again, equally vicious. Pyhrra (Sophitia's daughter) takes over this role in the fifth game, while Patroklos (Sophitia's son) will also be as bad if he's fighting Pyhrra.
Never fight Patroklos against Pyhrra if you don't want to be completely desensitised to the phrase "I'm sorry"... they will both apologise after pretty much every single attack, including their Critical Edges.
Rina from Fading Hearts apologizes to Ryou at the slightest chance of offence or disappointment she may have caused, especially when her behaviour becomes more evasive as the Visual Novel goes on.
Dragon Age II: During her introduction, Merrill apologizes for, in no order, asking Hawke's name, the standoffish behavior of the rest of her elfy clan, and verbal rambling. On the other hand, she's pretty firm about defending her decision to cut a deal with a demon for wizardly might and arcane knowledge.
I'm also sorry for listing examples from Web Originals and Web Comics
Asia Ellis from morphE. She is exceptionally polite, but does apologize way too much. When given a one-off chance to speak with her guardian, while in captivity, she spends half the conversation apologizing because her guardian is trying to get information on her location and she is unable to speak about it.
RWBY: Pyrrha Nikos has "I'm sorry!" as her practical Catchphrase. Even if she saves your life, she'll apologise.
We regret that we didn't mention Western Animation earlier
In Batman: Mystery of the Batwoman, Rocky (voiced by Kelly Ripa) is a klutzy scientist working for Wayne Enterprises that apologizes frequently, even if there is no reason. Though this is arguably a cover, making this a subversion of sorts.
Oh, sorry. I thought the open sign meant you were open, but I must have been mistaken.
You understand, don't you? You're not mad at me, are you? Please don't be mad at me!
Victor in Corpse Bride is this from time to time, especially when he's around Victoria, his bride in an arranged marriage. (He's never seen her before, and he's shy and clumsy, desperate to hold on to his manners.)
On Futurama, Fry does this while listening to his ex's story of what happened to her after he got frozen. He only apologizes because he feels bad for her and he does it at times when it's completely unnecessary.
Michelle: But it did give me a chance to think...
Fry: I'm sorry.
The Powerpuff Girls: Buttercup usually has trouble apologizing, let alone admit she did something wrong. But in "Moral Decay," after she accidentally clocks Bubbles and knocking out one of her baby teeth, she apologizes to Bubbles quite profusely.
During his first appearance in The Looney Tunes Show, Gossamer apologises to Daffy repeatedly, even when Daffy tells him to stop.
In Young Justice, Asami Koizumi, the show's take on Samurai, is like this, though it goes unnoticed because she speaks only Japanese. Sumimasen! ("excuse me," though it can also mean "thank you.") is practically her Catch Phrase.
Oh my, we forgot Real Life! Our apologies!
This is often associated with Canadians, with varying degrees of accuracy.
Conan O'Brien: What would you say is the most Canadian thing about you, about your personality?
Jessica Pare: I — well, I apologize to inanimate objects when I bump into them.
This a common trait among people with low self-esteem and lacking trust and/or social skills, as well as people who have been abused. It's a defense mechanism to prevent rejection, violence or hostility that one feels directed towards him/her, but also to dodge and avoid a conflict.
The apologizing when called out on it is often true because there's really nothing else to say to fair criticism.