I always thought that there was no real language class below type-3 grammars, but now I found the "Language identification in the limit" model which allows learning of pattern languages but not regular languages.
What exactly are pattern languages and what's their difference from regular languages?
Disclaimer: I am not familiar with language identification in the limit.
But I can say this: pattern languages are not “above” regular languages in the Chomsky hierarchy – in fact, they are not formal languages at all. Rather, they are structured but informal descriptions in natural language which don’t fit into the Chomsky hierarchy at all.
(Pattern languages are generalizations of the description texts used for the Design Patterns by the Gang of Four.)