The only one that comes to mind is Eric Matthews on Boy Meets World. In the beginning, his character was funny, smart, and cool. By the end of the series though, Eric was a half man child who was dumber than a box of rocks.
Can you think of any others?
Ross on Friends. (Yes, I liked the show. Sue me or look down on me, I don’t care.) He started out as a geeky, smart college professor and devolved into this insecure, quivering wreck of a neurotic.
Tim Baylis in Homicide changed drastically over the course of the show. He went from being a newb to being a standup cop, to being sexually confused (as an understatement), to going woo woo, to finally becoming a murderer.
And most of it was realistic, rather than just being a soap opera. (Most 
)
Wesley Crusher.
Went from being an annoying little kid, to being some sort of freakin other lifeform or somethin …
From TNG, Troi also changed. Went from annoying touchy/feely window dressing to commander status. Also was way less touchy/feely in the last half of TNG’s run.
Dr. Polaski. Went from being an annoying bitch of a doctor to falling to her death as an annoying LA lawyer.
Riker - beard.
Borg (in general) went from being the ultimate threat to being our friends (sort of) and sex partners.
Barclay. Went from a timid, but likable, sap to being a not so timid, and still likable man about town!
So, are you just talking about cases like the one outlined in the OP where the character is changed by random writer fiat (to play up aspects the viewers like, just because the writers don’t ‘get’ them, or whatever), or are characters who develop due to what’s happened to them in the series?
Wesley, in Buffy and then in Angel. Probably one of the most believable and richly written transformations in TV.
Hell, Angel as well. 
I came in to say this, because Wesley’s arc (starting on Buffy season 3 and then through all five seasons of Angel) is my favorite character arc ever, and he’s probably my favorite fictional character.
Cordelia changed a lot too, mostly for the better.
Damnit, I came in here to say Wesley!
fumes
Well, if I can’t have Wesley, I’ll go with Titus Pullo from Rome. From the loveable, amiable solider in the start to the . . . . monster, in the end.
Londo Mollari in Babylon 5. From a drunken buffoon to a semi-villain/semi-hero who sold himself for power without realizing it, to a doomed puppet Emperor.
Andy Sipowicz made a lot of progress on NYPD Blue. I can’t even begin to describe it. He was an angry, racist drunk when the show started. He was still a bit angry at the end, but he gained an understanding of himself and his world – mastered it, I think.
Greg Sanders, on CSI, went from bizarrely dressed, socially inept lab rat to snarky, geeky-cool field agent. I couldn’t decide which one I liked more. 
I’ve heard people on the board argue that Homer Simpson’s gotten a lot dumber and more violent as the series wears on…
Don’t forget Spike either.
Hot Lips Hoolahan in MAS*H. She stated out a bimbo with a military rod stuck up her butt. By the end of the show, she was a very sympathetic character. In the same show Klinger went from a minor character to the dress wearing nutjob to a coporal who was dealing with the military. For that matter Col. Henry Blake in that show. He started out alive.
The coach on Coach. He went from a very one-dementional dumb jock to a character who became, if not a rocket scientist, far from as stupid as he started out.
Wes, Angel, Spike…just about every character that survived the series had a tremendous growth curve. About the only one who was the person they began as was Giles, and I’d argue that one. 1st Season Giles would never have stood up to the Council the way 5th Season Giles did.
Trihs, don’t forget G’Kar. In the beginning a one-dimensional rebel, by the end a wise religious figure.
G’Kar and Londo basicly did 180 degree turns over the arc
I wasn’t much of a fan of the show but “All In The Family’s” Archie Bunker mellowed quite a bit in that show’s run, which eventually became “Archie Bunker’s Place”.
Though they never did explain how a guy who made numerous references to his Lebanese heritage and family got the name Max Klinger.
Frasier also mellowed over the course of 20 years, from total horse’s ass stock snob character to a still pompous but much deeper and more likable character by the end.
Carter went from a spoiled naive rich kid on ER to far nicer and less self-involved (especially once ER changed to “50% global issues or your money back” format).
In the early episodes of Happy Days, Ralph Malph was tough and street-wise. This persona then switched to Fonzie and Ralph became the class clown character.
The early episodes of Dukes of Hazzard were much darker and nowhere near as slapstick as the show eventually became. Roscoe was actually a very capable sheriff, just bitter (over losing his position/pension in Atlanta) and corrupt, while Boss was more of a small-town mob boss with ties to organized crime and also a womanizer. By the second season both were buffoons and by the third they were cartoon characters.
Kelly Bundy was always slutty on Married With Children but she wasn’t stupid in the original episodes.
Edith Bunker was a lot more sarcastic and wore pants in the early episodes of All in the Family.
In the first two seasons of Andy Griffith Aunt Bea had to come face to face with the effects of her longterm heroin addiction and emerged sadder but wiser, while most of Barney Fife’s blistering hatred of Catholics and foreigners was toned down. All of the episodes that came before Andy’s shooting of his wife, Svetlana, when he caught her in bed with a salesman from Mt. Pilot were dropped from syndication, as were the references to Opie’s sister Cassie who was kept locked in the basement due to what was euphemistically described as either a “glandular condition” or “antisocial tendencies”, and all of this was in order to make the show family friendlier. (Okay, I’ll admit- I made up the part about Barney hating foreigners.)