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This would require using a script. I've developed two tools that can help you do that. See:

Custom-made Adobe Scripts: Acrobat -- Create Copies with Unique Codes

or:

Custom-made Adobe Scripts: Acrobat -- Auto Increment a Field's Value

You can use the former to generate multiple copies of your file, each one with a unique value.

The latter can be used to change the increment the value of a field each time you print the file (by attaching the code to the document's Will Print event).