In any case, I get around the whole "grade grubbing" issue by having my policy of "extra" work spelled out, also up front on the syllabus. No one gets anything extra just for bugging me about it. Basically it amounts to allowing homework assignments to be re-done (once). A single quiz grade can be replaced by doing some fairly difficult extra problems. No makeups on the midterm and final, but a research project can be done to get up to 5 points on their final grade. Beyond that I hold office hours and happily spend them further helping students with anything they're stuck on.
Short of (provable) extenuating circumstances, that's about it. I even ask for a police report for students claiming "car accident" for things like missing a quiz. (In one case, a student told me there wasn't one because it was just a minor accident that scratched the side view mirror... I pointed out that an accident like that wouldn't have prevented them showing up to class, even if it was a few minutes late.)
My wife is a college psychology professor and is fairly strict with her grading scheme. She makes this clear on the first day of class and spells it out directly, in bold, on the syllabus.
Her rationale is this: The likelihood of you asking for extra credit, or excuses, or various "grade grubbing" activities is largely influenced by your socioeconomic background. Students who make special grading requests overwhelmingly skew white and female and are more likely to take place at institutions that take students from more affluent backgrounds. She doesn't want her grading inadvertently biased against males, non-whites, or less affluent students so her grades are final. Despite this inflexibility, her student evaluations are sky-high.