Relativistic mean-field calculations of multi-K hypernuclei are performed by adding K{sup -} mesons to particle-stable configurations of nucleons, {lambda} and {xi} hyperons. For a given hypernuclear core, the calculated K separation energy B{sub K} saturates with the number of K mesons for more than roughly 10 mesons, with B{sub K} bounded from above by 200 MeV. The associated baryonic densities saturate at values 2-3 times nuclear-matter density within a small region where the K-meson densities peak, similarly to what was found for multi-K nuclei. The calculations demonstrate that particle-stable multistrange (N,{lambda},{xi}) configurations are stable against strong-interaction conversions {lambda}{yields}NK and {xi}{yields}NKK, confirming and strengthening the conclusion that kaon condensation is unlikely to occur in strong-interaction self-bound strange hadronic matter.