Remove the neck and giblets from the chicken if included and place the chicken in a large stock pot.
Chop the carrots and all of the celery into 3 or 4 chunks each and add to the pot, along with onion, garlic, salt, pepper, oregano, and celery salt. Add enough water to fully cover the chicken and vegetables.
Bring the liquid to a boil, reduce the heat and cook at a low simmer for about 3 hours. The chicken should be very tender and easily come off the bone when done.
Remove chicken from the pot and set aside to cool a bit. Use forks (or your hands if cool enough) to pull off small chunks of meat. Remove all of the meat from the chicken and set aside. Discard carcass.
Strain the broth from the pot into a large bowl, discarding the vegetables or saving them for another use.
Add 6 cups of broth, the shredded chicken, and the rice to the pot and bring to a boil. Reduce to a simmer and cover. Cook for 15 minutes.
Add another cup of broth to the pot, stir, and cover. Simmer for 5 minutes more.
Continue to add broth, about 1/2 cup at a time, stirring and checking every 5 minutes until the rice is tender and cooked through. You'll need to taste test to check for doneness. Keep your heat fairly low to avoid burning the bottom of the rice.
Season with salt and pepper to taste. Serve hot.