Example of non accessible model categories

I don't know whether set-theoretic hypotheses are necessary to answer this question. But if we assume the negation of Vopěnka's principle, here is an example: by Example 6.12 of Adamek-Rosicky Locally presentable and accessible categories the locally presentable category $\bf Gra$ of graphs has a reflective subcategory that is not accessible, and by Proposition 3.5 of Salch The Bousfield localizations and colocalizations of the discrete model structure this reflector is the fibrant replacement functor of a model structure on $\bf Gra$.

An example which does not depend on set theory is the equivariant model structure on the category of maps of spaces by Emmanuel Farjoun.