What is the largest breed of domestic cat and how heavy are they when non-obese?
Maine Coon Cat.
According to this Ragdoll cats are the largest breed. (bolding mine)
I have two Maine Coon cats (brother and sister) named, Tibby (my namesake) and Libby. They are, indeed, quite large cats with great dispositions. My previous cat was also a Maine Coon (and also named Tibby). He had a perpetually tilted head from an equilibrium problem resulting from ingesting a neurotoxin laden blue tailed skink.
The reason I asked was that I have seen this shorthaired cat at the vet’s office, it was taller than most cats and it felt heavier than it looked.
Savannah Cats might be the biggest. As a Bengal owner I would kill to own one! ($2000 and up…)
This site offers lots of photos.
As lonesome loser said, the largest natural breed of domestic cat is Maine Coon. Friends of mine used to have one. Kit Kat was amazingly intelligent and trainable. He lived 17 or 18 years, IIRC.
While the yen for a big, big kitty cat is understandable, persons deciding whether to indulge that desire should remember that there are potential problems in coping with a critter that big, and of limited trainability, in a household setting. When all’s said and done though, as the old saying goes, “You pays your money and you takes your choice.”
I had an Applehead Siamese who, at 6 months was as tall on all fours, as my neighbor’s begals. He weighed 21 lbs, not a bit overweight, according to the vet. Of course, not anywhere close to the Maine Coon cats, but big, nonetheless.
(His name was Spot II after Downtown Spot Brown)
Is my post invisible or something? Despite a linked quote everyone just keeps rattling off “Maine Coon”. Given that the link is a site dedicated to that breed and may be biased I looked up Maine Coons and found this:
Given the above the Ragdoll has the Maine Coon beat for biggest breed of domesticated house cat.
Maine Coon: Big
I recall reading somewhere about a large cat breed called Ragdoll, but my Tibby can mop the floor with one of those. 
Whack-a-Mole how do you figure?
Maine Coon cat link
It looks like a tie from here.
My links made it close but not a tie.
My Maine Coon link:
“…males commonly reach 13 to 18 pounds, with females normally weighing about 9 to 12 pounds.”
My Ragdoll link:
“If a male is altered, it may reach the weight of 20 pounds or more. Females are smaller and look more feminine. They usually weigh about 8-15 pounds.”
According to those the Ragdoll edges out the Maine Coon for size. Granted there will be variations and different sites my list different ranges (as yours seems to have) but the Ragdoll site seems content to grab the title for “largest domestic cat” breed. While I suppose they could do that with little regard for the truth I have yet to come across something that says the same about Maine Coons.
Actually, it looks like that site was copied from this one which makes the weaker claim that Ragdolls are “one of the largest cat breeds”.
The longest cat is a Maine Coon, weighing in at 35 lbs; he’s 4’ long from nose to tail. (Guinness apparently doesn’t keep world records of heaviest animals because that’d encourage people to fatten up their pets.)
So Ragdolls may be larger in general as a breed; I’m having trouble finding verification for this. But Maine Coons have the largest member of a breed on record.
Daniel
Norwegian forest cats are also very large. Males can be between 13 and 22 pounds. They look a lot like Maine Coons. Just thought I’d throw that out there.
I think it may be that many people regard Ragdoll cats as being more than a little weird, and not ‘normal’ domestic cats at all.
Doesn’t sound much like a proper cat to me, but then again, would make a nice change from being mauled by my mother’s cats.
I don’t think the aforementioned Kit Kat ever weighed more than 20 pounds in his prime, but he was never fat. Although well-fed, he was always very “rangy”; maybe he was concerned to stay at his best “fighting weight”? 
I do assure you that he was by far the largest specimen of Felis domesticus I’ve ever seen. And I’ve seen an awful lot of cats in my 60+ years.
He’d been the recipient of a bungled neutering before acquisition by the owners I knew. That is, he was unable to breed, but his territorial instincts were unimpaired. He was the undefeated feline ruler of his domain of several city blocks, but he was also an extremely gentle companion/playmate to his people and their guests.
Did anybody say Maine Coon yet? 
Just kidding…
My Bengal is not unusual, male/fixed (ex-male I call it), in that he is 19lbs of lean muscle. I’m not sure how far removed he is from the SE Asian Leopard cat he is related to, but I know it has to be at least several generations to be listed and is likely many more than that. He’s very domestic. The idea of Savannah cats bothers me, as I know Bengals are a bit hard to take care of than many cats (just a bit more energetic and prone to behavioral problems (which is how I got my cat cheap–the breeder no longer could deal with him!)) and I worry that a larger version will just get dumped that often more. I’d hate to see a pack of feral 30-40lb hybrid cats in the city. (Toddlers beware!)
How about cats with the largest heads? Years ago, I recall reading about feral cats in the New Jersey Pine Barrens that had exceptionally big, mutated heads – as well as particularly unpleasant personalities. I can’t find a cite for this. Urban legend? Perhaps, but I don’t think so.
Every time I see the title of this thread I think of Chuck Connors pulling up on a motor cycle and saying something about ‘a special breed of cat’. (Thrill Seekers)