Every now and then, I burn a pot while cooking. It doesn't happen too often and usually, with a bit of elbow grease I can scrub it off. However, one time I made a very large pot of soup in my biggest and nicest stock pot. I had it on low to keep it warm during a party and unfortunately burned the whole batch. The bottom of the pot was badly burned. It had a very thick, black and incredibly hard layer of burnt soup on the bottom! I tried everything I could think of to get my pot clean. I boiled water in it, cleaned it with tough scrubbers and cleaners, soaked it for a couple days. Although the burnt residue came off a little, I still had a very burnt layer on the pan that I was unable to remove. I even boiled tin foil in the pot because I heard that worked...it didn't. If it had been one of my cheaper pans/pots I probably would have just thrown it away. However, I wanted to keep this pot and wanted to get it clean.
I asked people for advice and someone (maybe my sister?) suggested boiling a dryer sheet in the pot. Since I don't have any dryer sheets (I use dryer balls) I finally asked my friend for one and tried it out. Boiling the dryer sheet made my house smell very strongly of dryer sheet but it totally worked! I was elated. I had no problem scrubbing out ALL of the remaining residue after using the dryer sheet. It still took a little bit of elbow grease but I could immediately tell a difference and it actually was able to come off this time. So, if you have a pot or pan you need to salvage, try a dryer sheet. You might also try soaking it in water without boiling it. There's a chance that might work without having to breathe in all the dryer sheet fumes.
Do you have any other tips for cleaning difficult pots and pans?