From another viewpoint, all human language is only at the finite-state level - a finite state automaton can recognise a language from any level in the Chomsky hierarchy provided you constrain all sentences to a finite maximum length, which of course you can - no human utterance will ever be longer than a googolplex symbols, since there aren’t enough atoms in the observable universe to encode such an utterance

Really the way people use the Chomsky hierarchy in practice (both in linguistics and computer science) is “let’s agree to ignore the finitude of language” - a move borrowed from classical mathematics

This is why the classical notion of competence and performance in linguistics is important. We describe programming languages as being Turing-complete even if every computer is always in practice finite because, in principle, it could be run on a computer with more memory or whatever. Likwise, it seems that language is bounded by language external facts about memory not intrinsic facts about how language is processed.