My top ten (after Archie Bunker):
Harry Solomon (3rd Rock From the Sun)
Rose Nyland (Golden Girls)
Felix Unger
Edmund Blackadder
Nigel Wick (Drew Carey Show)
Dr. Sheldon Cooper (Big Bang Theory)
Ted Baxter
Cpl. Randolph Agarn (F Troop)
Mork from Ork
Phyllis Lindstrom
Oh, yes! How could I forget Mr. Wick? 
A few more that occurred to me:
Exidor (Mork and Mindy) - small part, but made a big impact. A couple of the scenes with him and Mork made me almost hurt myself laughing.
Father Dougal McGuire (Father Ted)
Father Jack Hackett (Father Ted)
Mary Hartman Mary Hartman
Who is Louie DePalma?
Seriously, I’d like to know. I suppose I could wiki it, but I guess my real question is, why was he so awesome? It’s the only name on this thread that I’ve completely never heard of.
It’s also interesting to me, reading this, how age centric it is…if there was a bell curve of the names mentioned on this thread vis-a-vis when the shows were on, the peak would be about 1977 or so…I wonder if that’s because there’s a peak sit-com-watching time in one’s life, or if because we always like the shows of our youth the best, or if because the 70s really were a golden age of the sitcom while other genres were more dominant in other eras…
Also, I wonder what people would name as the greatest purely American sitcom character? Archie Bunker seems to be the consensus, but as has been pointed out, he’s a rip off of a British show. Maybe Honeymooners? I haven’t seen it much, as it’s not much to my taste, but it does seem to be the ur-show, the prototype.
Oh, and shout-out the the Brits — if all these other shows are considered the best of the 70s British sitcoms, why in god’s name are y’all constantly running re-runs of Only Fools and Horses? Bleck. Although I do have a secret fondness for Are You Being Served. It’s a bit like, say, Bird’s Eye custard or Campbell’s soup …the idea has an appeal the actual experience often lacks.
My first thought was Hawkeye, but I wasn’t sure if MAS*H is really a sitcom (laugh track or no laugh track.) But since it’s in, Radar deserves a nod too.
I’m not a fan of the Office in no small part because it already seems dated to me. Part of it’s the camera work and part of it’s set up, but it feels to me very reflective of a specific type of show and a specific type of situation from a specific period in time - late 90’s office life back when Dilbert was still fresh and stuff shot informally on handheld digital cameras was an exciting new approach. Is Dwight more universal and evergreen than Dilbert? I would doubt it. But as I say, I’m not a fan and I don’t watch often.
Without looking at any of the other entries, I’m going to vote for Hank Kingsley (Jeffrey Tambor) from The Larry Sanders Show. The runners-up are so pale in comparison, they are barely worth mentioning (Homer Simpson, Elaine Benes, George Michael Bluth…). Hank is a rat bastard. Hank is a tragic figure. Hank is hilarious.
Balki Bartokomous
He was Danny DeVito in Taxi.
I think the 1960s was more of a Golden Age of sitcoms. Gilligan’s Island, I Dream of Jeannie, Dick Van Dyke, many others. Although I suppose an argument could be made for the 1950s, too. But whoever THE greatest character in sitcom history is, I suppose it should be one on a program that has stood the test of time and become part of American culture. As much as I like George Burns and Gracie Allen, I have to admit it’s only older people who even knew they were ever in a sitcom to begin with. But everyone knows Gilligan or Hawkeye or Jeannie.
Gracie Allen, it has to be. Every time I think I know where she is going, she takes a different tack.
I just realized, Gracie is the other side of Mrs. Cake. Watch the show, especially any in which she has dealings with public officials.
Also they do classic sit com plots, but intertwined with them is the absolute madness of Gracie.
I’m not even sure why, but seeing other people who appreciate Gracie’s humor just warms my heart. She deserves to be remembered and celebrated forever!
I think Marie is amazing. Doris Roberts makes her believable, loathsome, sympathetic, and hilarious. She’s my favorite part of that show, by far. But I don’t know if she’s the greatest character in sitcom history.
I have to define “sitcom history” as “the shows I happen to have seen.” I’m 24 and didn’t grow up with a lot of the shows people have been talking about in this thread. I have a very cursory knowledge of shows like Taxi, but I’ve never heard of half the characters in this thread. It’s sad, really. Maybe I’ll enlist Netflix to fight my ignorance.
I think “greatest” has to be more than just a personal favorite (for me, that’s Larry David, followed by Jerri Blank). The greatest character has to be embodied by a gifted comedian and should be an iconic representative from a truly great, influential sitcom.
With those conditions, my pick is Kramer. Bill McNeal is my runner-up.
Hey, kid… a lot of us are talking about shows that were on before we were alive too… I think Gracie Allen died when I was still in diapers…
Now, get off my lawn! 
Although not the funniest by any stretch, my vote for a typical middle America character would have to be Chester A. Riley. Pounded rivets all day, carried a lunchbox and had a white picket fence.
I don’t know if I missed it… but I can’t believe nobody has mentioned
Dan Fielding (John Laroquette) from Night Court
Not only was he funny, but Laroquette was spot on. He asked to have his name removed from consideration because he won Best Actor Emmy 4 straight years. (cite)
Also- I see somebody mentioned Liz Lemon- but I would put Jack Donaghey as the truely great character in that sitcom.
Sir Humphrey Applebee (Yes Minister).
But I think his role in the John Larroquette Show was better (the writing was better, too), where he played a recovered alcoholic, night manager of a St. Louis bus terminal. The episode where the ghost/Jesus/Willie Nelson appears on the wall is a classic every skeptic should watch.