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Choosing pot size for the snow-melting task doesn't make much sense.  Pot size should mostly be a function of what weight you're willing to carry and how much space you have.  That's obviously quite different when you're backpacking as opposed to car camping, for example.

A better minimum pot size is to hold "cup" of whatever you can drink stuff like hot chocolate from, however much water you need for one packet of soup, etc.  Think about what you really need to heat as one whole pot, and use that size.

Melting snow can be done in any size pot.  Snow is much less dense than liquid water, so one full pot of snow isn't going to make much water.  However, that doesn't matter.  As snow melts, keep filling the pot again until you have a whole pot of liquid water.  Repeat as needed.  There is little advantage to fewer large batches versus more small batches.  It's a continuous process either way.  The only difference is how often you empty the pot into a canteen or whatever.