What to do about "grade grubbers?"
I usually say that my rule for regrades is like the NFL's rule for replay challenges: there must be "indisputable visual evidence" that the original grade was incorrect.  For instance, scores were added incorrectly, or a correct answer was marked wrong.

If the score was a "judgment call" (for example, I deducted two points for some error and the student thinks it only deserved one point deducted), I won't change the grade, unless I did something really egregious.

I've seen the policy "We regrade the entire assignment and your grade could go up or down" but it makes me a little uncomfortable.  It suggests an element of randomness or caprice in the grading, which I don't want to promote, especially for younger or weaker students who may already feel like their grade is random.  I don't think students should feel like they have to roll the dice to get a genuine error corrected.