The ratio of the plasma pressure to the magnetic field pressure (or of their energy densities) which is known as the plasma parameter "beta" (β) has important implications to the propagation of energetic particles and the interaction of the solar wind with planetary magnetospheres. Although in the scientific literature the contribution of the superthermal particles to the plasma pressure is generally assumed negligible, we deduced, by analyzing energetic particles and magnetic field measurements recorded by the Ulysses spacecraft, that in a series of events, the energy density contained in the superthermal tail of the particle distribution is comparable to or even higher than the energy density of the magnetic field, creating conditions of high-beta plasma. More explicitly, in this paper we analyze Ulysses/HI-SCALE measurements of the energy density ratio (parameter β ep ) of the energetic ions' (20 keV to ~5 MeV) to the magnetic field's in order to find occurrences of high-beta (β ep >1) superthermal plasma conditions in the environment of the Jovian magnetosphere, which is an interesting plasma laboratory and an important source of emissions in our solar system. In particular, we examine high-beta ion events close to Jupiter's bow shock, which are produced by two processes: (a) bow shock ion acceleration and (b) ion leakage from the magnetosphere.