The next one is called 

<a href="http://ncatlab.org/nlab/show/modification">modification</a>.

Modifications between ordinary natural transformations between functors between categories are trivial, but they exist between 

<a href="http://ncatlab.org/nlab/show/lax+natural+transformation"> lax natural transformation</a>s

between 2-functors between 2-categories.

The next one after that is called _perturbation_ : a perturbation goes between pseudonatural transformations between 3-functors between 3-categories.

Beyond that, no established terms exists. Instead one starts numbering things and speaks of 

<a href="http://ncatlab.org/nlab/show/transfor">transfors</a>.

n-Functors are 0-transfors.

Transformations are 1-transfors.

Modifications are 2-transfors.

Perturbations are 3-transfors.

And so on.