Usually the receptacle is the first to go. Obviously this would be a bigger problem the more you have plugged in to one recep. The metal springs that hold the prongs of the attachment plug in the recep lose their springiness with repetitive or excessive heating. Then the attchment plug is lose which causes arcing. Beyond arcing being very hard on some electrical components(LED drivers come to mind), it also causes corrosion, and you guessed it, more heat then this happens if it’s not replaced soon enough.
This was caused by a large electric wall furnace at my buddies house. He got stuck on the circuit when he tried to unplug it. His cousin had to push him off with a broom stick.
Breakers will also wear over time in a similar fashion to outlets. When they wear out they start tripping a lot. If not replaced, they will eventually stop tripping and if a fault occurs there is nothing to stop the overcurrent. Thats the really scarey one. Arc faults can do some serious damage. In a residential application the wire would probably melt before you get a serious Arc flash or blast, but you still have one hell of a fire on your hands. One of the first jobs we got after I started my apprenticeship was replacing a 2000A main in a high rise office building. The kicker is there were several federal call centers in the building and they couldn’t be rerouted in time for emergency repairs so we had to get special dispensation from the state to do it live. My boss made me do the Arc fault calc to impress on me just how dangerous it was. He made me hold the shepherds hook a few feet away while they did the job but the reality is we all would be ash before I pull anyone off. I damn near s#@t my spaceman suit a few times. The boss even called me the night before to tell me not to smoke any pot. Since then I realized that no matter what your doing electricity is dangerous if your not careful and well informed. If you have any more questions you can hit me up my journal whenever and I’ll do my best to answer them.