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I didn't get into a clinic and am graduating next semester, so this is my last chance at taking a clinic. (I applied to a couple, but I really wanted to get into my first choice.) Is there any way to appeal? Formal process or is it just up the profs? Should I cry to them or what? Advice?

They are all prof pick, I would honestly just go to them and talk to them about your situation. May not work, but sitting down face to face with them can do nothing but show you are serious about your desire to participate in a clinic before you graduate.