@quantensalat
Probably not useful, don't lock me up. But if you would settle for abelian gerbes, and don't mind working in local coordinate patches (and Cech cohomology) ...
Then gerbes are to line bundles what line bundles are to (nowhere-zero) functions.
When you use functions as transition functions, you get a line bundle. When you use line bundles as transition functions, you get a #gerbe .
And you can keep going to higher order, if for instance you want to get to some differential form of higher degree that you want to express as the "curvature" of some "n-gerbe".
Only works for abelian though, as best as I know.
