This topic comes up all the time and I agree with most of what others have said, but not all of it. I think self-studying APs can occasionally help strengthen an application, but I think the impact is specific. Case in point, my D self-studied AP psych as a junior, took the exam and got a 5. She made it very clear on her app that she is most likely going to major in Psychology, and her school doesn’t offer the course. (If the school had offered the course, there would be no reason to self-study.) My D did very well in the admissions process and I definitely think that showing the discipline to seriously study a subject and get the top grade on an exam did nothing but help her application. I am sure it was a minor consideration, but I find it hard to believe it was ignored.
I also totally disagree with @VickiSoCal that AP psych is not highly regarded. My D in fact contacted a professor at her (highly regarded) college and asked if she should take Psych 101, explaining the circumstances. The professor was majorly impressed. Not everyone wants to study physics and chem.
@skieurope , do you not think that sometimes self studying CAN be helpful? 