If one were cooking a pot of pasta or a soup and later found a bug or many bugs in the pasta or soup, would the pot need to be re-kashered after it was thoroughly cleaned?
No need to kasher once the bugs have been cleaned out, because the halachah is that bugs are נותן טעם לפגם, they impart an unpleasant taste [that does not enhance the flavor of other foods]. Source: Yoreh Deah siman 104.
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Your average bug is probably not much bigger than a half a centimeter. That means its volume is 0.125 cm^3. Assuming a bug is roughly the same density as water, a bug should weigh around 125mg. 60 times the weight of a bug would be roughly 7.5 grams. Most pots contain more food then that.
I did it in weight because it is easier to appreciate weight measurements, it would come to roughly the same result if one calculated in volume, a pot 60 times .125 cm^3 would be 7.5 cm^3. That is definitely smaller than your average pot.
In other words, there is a good chance that the bug would be batul in the pot.