Owing　to　the　presence　of　exceptional　points　(EPs),　non-Hermitian　(NH)　systems　can　display　intriguing　topological　phenomena　without　Hermitian　analogs.　However,　experimental　characterizations　of　exceptional　topological　invariants　have　been　restricted　to　second-order　EPs　(EP2s)　in　classical　or　semiclassical　systems.　We　here　propose　an　NH　multi-mode　system　with　higher-order　EPs,　each　of　which　is　underlain　by　a　multifold-degenerate　multipartite　entangled　eigenstate.　We　implement　the　NH　model　by　controllably　coupling　a　Josephson-junction-based　electronic　mode　to　two　microwave　resonators.　We　experimentally　quantify　the　topological　invariant　for　an　EP3,　by　mapping　out　the　complex　eigenspectra　of　the　tripartite　system　along　a　loop　surrounding　this　EP3　in　the　parameter　space.　The　nonclassicality　of　the　realized　topology　is　manifested　by　the　observed　quantum　correlations　in　the　corresponding　eigenstates.　Our　results　extend　research　of　exceptional　topology　to　fully　quantum-mechanical　models　with　multipartite　entangled　eigenstates.　Exceptional　topology　associated　with　higher-order　exceptional　points　has　not　been　completely　characterised　in　quantum　systems.　Here,　the　authors　fill　this　gap　performing　full　quantum　state　tomography　in　a　system　composed　of　a　superconducting　qubit　coupled　to　two　cavities,　one　lossless　and　one　lossy,　featuring　third-order　exceptional　points.