How long can cooked hamburger meat stay fresh at room temperature?

The answer to this question is completely subjective. In my opinion, if the meat looks slimy and smells bad, throw it out. However, if it's only a bit rancid, you could cook it thoroughly on a really high heat and then hide the flavor with curry or chili. This will kill off most of the bacteria and your stomach acid will do the rest. This is how they prepare meat in the tropics. You should take into consideration that the meat you buy in supermarkets is at least several weeks old. It will have been hung after slaughter (a bit of decomposition enhances the flavor), probably frozen and then defrosted before sold as fresh. Refrigeration slows down the decomposition so depending on how old the meat was when you bought it you could be looking at anything from 3 to 30 days. Use your nose. If it smells rotten, it is. Remember you can safely ignore the sell-by date as supermarkets can take the meat off the shelf and repackage it up to 3 times before you buy it. My advice is to buy from a local butcher. Avoid supermarkets. Ask the butcher where they get their meat. If they buy from a wholesaler, go to a decent local butcher.