There are such natural examples, such as the very recent ICALP'20 contribution on "unambiguous finite and Pushdown Parikh automata" (paper not yet available), or the related RCM class by Massazza and Castiglione. In both cases the generating function of these language classes are holonomic and moreover contain examples which are not algebraic.
EDIT: The ICALP'20 paper is now available.