What to do about "grade grubbers?" An annoying aspect of working with a "strict" grading system (e.g., 90%+ = A, 85% = A-, 80% = B+, etc.) is what I call the "grade grubbing" phenomenon: the people who feel compelled to raise complaints about the grading because they didn't get the final score they want—but have no real argument in support of a higher grade. This is not about legitimate requests to reconsider because of a mistake but instead students searching for any reason why they deserve the X points they need to get a better grade. It’s exemplified by the kind of argument that begins “I know it’s wrong but . . . .”

I do not have the power to change the grading structure—that is imposed from higher up. 

Are there any satisfactory methods of discouraging such behavior? I don't want to stop people with legitimate issues from asking for regrades (mistakes happen!), but I would like to avoid having to deal with the student who tries every which way to get the few points they need to move up a level.
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.