“What do different
\fontdimen<num> mean” lists but does not mention how these three font dimensions are used. I could not find this information anywhere. All I could figure out was that changing s14 affects the usual superscript position, but could not find out how s13 and s15 were used. Even “Redefine the default depth of superscript and subscript document-wide” does not bother to change s15. So what are they really?
Edit: I've provided my own rough answer after seeing cfr's examples and fiddling with many more.