Relativistic mean field calculations of multi-$bar{K}$ hypernuclei are performed by adding $K^-$ mesons to particle-stable configurations of nucleons, $Lambda$ and $Xi$ hyperons. For a given hypernuclear core, the calculated $bar{K}$ separation energy $B_{bar{K}}$ saturates with the number of $bar{K}$ mesons for more than roughly 10 mesons, with $B_{bar{K}}$ 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 $bar{K}$-meson densities peak, similarly to what was found for multi-$bar{K}$ nuclei. The calculations demonstrate that particle-stable multistrange ${N,Lambda,Xi }$ configurations are stable against strong-interaction
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