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Say I fixed some stew that's been in the refrigerator for around three days. Will putting that stew in a pot, heating it up to around 170F (77C), and letting it simmer for a while make it last longer in the refrigerator? And if so, then can this be done (nearly) indefinitely?

I think the answer is yes, because whatever microbes have had an opportunity to take root in the food will have been killed by the high temperatures, but I am not sure about whatever biproducts that might remain (e.g. toxic chemicals exuded by the microbes as part of their metabolism or something).

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Yes, it will "make it last longer in the fridge" for a certain definition of "last longer": it will not mould nor infect other foodstuffs in the fridge if it doesn't come into direct contact with any of them.

Is it still safe to eat after a few of these re-heatings? Very probably not!

Elaboration: More then half of the number of cells in your body are actually bacteria; bacteria don't generally kill you but the toxins they release is what kills you. E.g. Botulinum Toxinum D (one of the toxins released during Botulism) has an LD50 of 0.4 ng/kg, which means that you need about 500g (a pound) to kill 99.9% of all human beings on the entire planet...

(And that is for just a few re-heatings; for "nearly indefinitely" we need to bring extremophiles in the equation...)

Related, but not identical question

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    Here's a more generalized version of that same question: cooking.stackexchange.com/questions/16872/…. It's also worth mentioning that quality and nutritional value will degrade with each re-heating, so this definitely can't be done any more than a few times for multiple reasons. – logophobe Apr 16 '15 at 16:46

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