I love mexican-inspired chicken soup:
Simmer the chicken bones in salted water with some chopped up carrot and celery along with a halved onion (chop off top and cut in half through the root, then remove the skin so that the two onion halves stay intact and don't fall apart in the soup), a bay leaf and maybe a couple smashed garlic cloves. If you want more meat, I throw in a full thigh or two (since they are cheap). Skim the foam and let it go for 1-2 hours or however long you can afford. Remove the bones, mushy halved onion and thighs and set thighs aside. You shouldn't have to use any soup base or pre-made stock for this recipe.
Add a can of black beans (low sodium or, even better, dried beans you've cooked yourself), a can of diced tomatoes and a chopped up chipotle pepper (the kind that come in cans in adobo sauce). Shred the thigh meat and add back. You can add anything else you like - corn, potato, other vegetables, leftover rice, whatever else is lying around. I also usually add some mexican oregano and a dash of garlic powder and cumin.
Simmer until whatever ingredients you've added are cooked through. Salt and pepper to taste.
Here's what really gives it flavor - serve with a bunch of fresh toppings. Chopped up fresh cilantro and red onion, crumbed cojita cheese (cheap at your local mexican market), a handful of crushed up tortilla chips, avocado, and some diced fresh jalapeno if you like it spicy. Finish with a squeeze of lime juice for some acid to balance everything out.
Another tip - if you get into oven roasting a whole chicken or thighs/legs for the family, the carcass/bones make an amazing starting point for soups.