Where do transformational grammars stand in the Chomsky Hierarchy?
I was taught in linguistics class that Chomsky abandoned his original arbitrary transformations after they were found to produce Turing completeness. See e.g. the Oxford Handbook of Linguistic Interfaces, p. 543; I am unable to locate the original reference.

He subsequently developed a formalism with more restricted transformations; the version I was taught was called Government and Binding Theory.

Some details are in Wikipedia under Transformational grammar.

While searching for references, I ran into a highly readable article on this matter by Geoffrey Pullum, which rejects Chomsky's original notion of transformations for purely linguistic reasons.