Why are most cats mixed breed?
First, obtaining a cat with papers is a lot more expensive. The lowest I've seen (posted price on a website, I've never seriously talked to a breeder) has been US $600 and some rarer breeds can be in the thousands.
By contrast, a DSH/DLH can be obtained from a rescue or shelter for between $5-150. Many people adopt kittens from people in their neighborhoods or a friend of a friend (free) when a feral or unspayed outdoor cat has a litter. Sometimes an amazing friendly cat randomly shows up at your house (this happened to us).
Additionally, cat breeds aren't differentiated as much as dog breeds. For example, cats are all roughly the same size, so someone isn't going to go look for a particular breed because they don't have the strength to handle a 120 pound cat on a leash.
Cats haven't been bred for particular TASKS like dogs so they don't have huge differences in personality like dogs. A toy type dog has a very different personality than a herding type dog, because they had different jobs. That differentiation has never happened in cats. Cats had one task for thousands of years (kill vermin), and more recently have a second task (companionship).
So if a prospective cat owner does not need to pay the purebred premium to ensure certain traits (because there's not that much differentiation in cat breeds) then why would they?
Practices of Cat Breeders
Second, when someone starts to talk to a breeder about obtaining a purebred cat, one of the things the breeder will ask is what is the purpose of this cat?
If the prospective owner only wants a companion, often breeders will provide a "pet quality" cat. This is a cat that does not represent an improvement in the breed (i.e. its traits are further from the breed standard than its parents). Allowing this cat to reproduce would degrade the breed, so it is desexed (either by the breeder or required by the contract that the new owner have it done).
Since the cat is desexed, these purebred traits are not going to be accidentally mixed with the large population of DSH/DLH (or other purebreds), so you're not going to see a half Ragdoll half DSH.
To obtain a purebred cat who has not been desexed, a person generally has to show themselves to be very familiar with the breed and dedicated to the hobby. One common way of doing this is purchasing a desexed cat and showing it for a year or three. One site explains that you can plan on spending about $2000 for the purchase of a desexed cat to show plus fees and expenses to show the cat and network with established breeders.
Once you've done this, breeders who know you may be willing to sell you an unaltered cat, for another couple thousand dollars. It's unlikely at this point that someone who had taken these steps would then allow their purebred cat to indiscriminately breed with random other cats. So, again, a half-breed (unpapered) cat is unlikely to ever be created.
The cats born from a purebred cat with papers are almost exclusively going to be other purebred cats with papers.
But the rescue said my cat is a particular breed!
With the advent of the internet and aggregation web sites like Petfinder, rescue organizations have to market their pets more aggressively. One way they do this is by assigning breeds to adoptable pets.
These breed assignments should be understood to mean "this cat looks like BREED" or "this cat appears to have traits common to BREED". How well these assignments are made depends on the knowledge of the rescue staff and how aggressive they are trying to be in their marketing.
Some rescues will label all gray, non-striped cats "Russian Blue", yet you can click through hundreds of rescue "Russian Blue" cats and not find a single one with anything approaching the bone structure of a Russian Blue, many even lack the eye color (which is easier to see even if you aren't familiar with how Russian Blues are built). In the context of rescue, Russian Blue means "gray with no stripes or white spots".
The exception to this is a breed specific rescue. These rescues are familiar with the breed and will only accept cats of that breed. They make this determination either through papers or by extensive familiarity with the breed (it varies according to each particular rescue). Even though I've gotten most of my cats from the Siamese Rescue Center, that doesn't mean that they're purebred Siamese (they are definitely not). It means that these cats have been pre-screened to have many of the traits of Siamese.
Exceptions to the Above
Despite the best practices of breeders (as explained above), there are some number of unaltered purebred cats that have been obtained by unscrupulous breeders (aka 'backyard breeders'). This is more likely to happen in more common (popular) breeds that have a large number of breeders.
If there's only <10 breeders of a particular breed in the US, they probably all know each other or at least know of each other. It's more difficult to get an unaltered cat of that type of breed than a breed that has a bunch of breeders and while a breeder knows a lot of other breeders, they don't expect to know everyone.
Additionally, there has been more time for these accidents (letting an unaltered cat go to an unscrupulous breeder) to happen in older breeds that have been around for decades than in younger breeds that have only been around a few years.
This means that these specific breed traits can be found in the common mixed breed population. The big three breeds that I've seen this happen to on large scale are
- Maine Coon
Is this particular cat a Maine Coon, Ragdoll, or Ragamuffin?
If you don't have papers, most likely it's none of the above. It's a mixed breed (or Domestic Short/Medium/Long Hair).
Could this cat be the result of one of the exceptions?
Maybe. Consider first the age of the breed, then the popularity (I couldn't find number of cats registered each year, nor even number of breeders from TICA, but CFA lists breeders so we'll use that as a stand-in for popularity).
So, based on both age and popularity, the breed most likely to be an exception as explained above is the Maine Coon. So if the cat has a bunch of the traits that these breeds have in common and you had to pick one of these breeds as likely ancestry, I'd pick Maine Coon.