I used the recipe from Budget Bytes ($5.87 recipe / $0.73 serving).
1 medium yellow onion
2 stalks celery
2 medium carrots
1 Tbsp minced garlic
3 lbs. potatoes
2 Tbsp chicken base
6 cups water
1/4 tsp cracked pepper
2 cups milk
1/4 cup all-purpose flour
1 tsp salt
STEP 1: Finely dice the celery and onion. Peel the carrots and then grate them on a cheese grater. Mince the garlic or use pre-minced from a jar. Add all of this to the slow cooker. Clean the potatoes well, cut into one inch cubes or smaller, and add them to the slow cooker.
STEP 2: Prepare 6 cups of chicken broth by dissolving the chicken base in 6 cups of water (or use homemade or store bought broth). Add to the slow cooker. Add some freshly cracked black pepper (about a 1/4 tsp). Secure the lid on the cooker and cook on high for approximately 8 hours (give or take an hour).
STEP 3: Open the slow cooker and test the potatoes for tenderness. They should be very soft. The onions and celery should also be very soft and transparent. Whisk 1/4 cup of flour into 2 cups of milk and then stir that mixture into the soup. Secure the lid once more and let cook on high for another 30 minutes.
STEP 4: After 30 minutes, the soup should once again be bubbling. It needs to be bubbling for the flour to have it's full thickening power. Use a potato masher, immersion blender, or a hand mixer to "mash" the potatoes. Or, you can transfer half of the soup to a blender and puree (be careful, it's HOT) until smooth and then stir it back in to the rest. Taste the soup and add salt as needed (about one teaspoon). Serve hot.
Above is exactly as Beth has it written. Now, here's what I did (i.e. what you should not do):
STEP 1: Food process the celery, onion, and one giant carrot. While this is ok, the giant carrot is not. Stick to 2 medium carrots or 1/2 a giant carrot. Add the mirepoix to the slow cooker. Add jarred minced garlic. Jarred minced garlic is my friend! If you are a garlic snob or like to mince tiny objects, be my guest. Clean and cut the potatoes as directed above. Add to the slow cooker.
STEP 2: Follow the above steps using homemade broth. Only the day before, make chicken broth and freeze it until solid in a 2-quart plastic pitcher. Forget to set the pitcher in the fridge in the morning. Say foul word while digging at the broth in the pitcher. Be brilliant and run hot water over pitcher sides until broth can slip free. Dump frozen broth in slow cooker.
Realize that this will NOT work, dig out largest bowl in kitchen, spoon potatoes and mirepoix into bowl. Settle frozen chunk of brothy goodness into bottom of slow cooker. Dump potatoes and mirepoix into slow cooker, onto self, and counter. Scoop escaped ingredients back into slow cooker. Set cooker on high. Cook for 1 hour, check frozen broth. Break apart with fork. Cook for remaining 7ish hours.
STEP 3: Test potatoes for tenderness. Stir a little. Burn hand with hot broth. Follow remaining directions above.
STEP 3a: Decide to boil remainder of 5 lb bag of potatoes to add to soup. Set up dutch oven with water, salt, and chopped potatoes. When done, add potatoes to slow cooker. Watch in horror as liquid level rises and nothing bubbles. Dump potato water from dutch oven. Scoop off about 4 cups liquid from slow cooker into dutch oven. Set stove to high and ignore until boiling over. Burn self again while saving broth. Clean up mess.
STEP 4: After about 1 hour, give up seeing broth thicken. Scoop potatoes (about half) into food processor. Hit that pulse button, baby! Instant mashed potatoes. Add broth until liquidy. Dump into dutch oven. Decide to scoop remaining potatoes into food processor. This is a bad idea. Consider that potatoes are starch and starch loves soaking up liquid. Here is where you DO NOT want to do what I did. If you REALLY want all the potatoes mashed, just fork mash the whole thing. You will avoid the following frustration.
STEP 5: An entirely NEW step! Scoop all of the mashed potatoes/soup into slow cooker. Rearrange fridge to fit slow cooker insert. Go to work. Come home. Scoop about 4 cups of mashed potato soup into dutch oven. Heat. Remember to turn on high so that the molten potatoes will bubble up onto your hands and arms (do I every learn?) as you race to turn down the heat. Realize that your "soup" is now in fact mash potatoes. Break out the carton of broth -- because the last of your homemade stuff is still frozen, of course -- and start adding until you get the right consistency. Recognize that your soup is never really ever going to be soup because those potatoes are broth sucking fiends now.
This is actually very tasty and super simple to make. I just like making my life difficult. It adds excitement...and burns.