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High entropy alloys (HEA) have unique properties including the potential to be radiation tolerant. These materials with extreme disorder could resist damage because disorder, stabilized by entropy, is the equilibrium thermodynamic state. Disorder also reduces electron and phonon conductivity keeping the damage energy longer at the deposition locations, eventually favoring defect recombination. In the short time-scales related to thermal spikes induced by collision cascades, phonons become the relevant energy carrier. In this work, we perform a systematic study of phonon thermal conductivity in multiple component solid solutions represented by Lennard-Jones (LJ) potentials. We explore the conditions that minimize phonon mean free path via extreme alloy complexity, by varying the composition and the elements (differing in mass, atomic radii, and cohesive energy). We show that alloy complexity can be tailored to modify the scattering mechanisms that control energy transport in the phonon subsystem. Our analysis provides a qualitative guidance for the selection criteria used in the design of HEA alloys with low phonon thermal conductivity. [ABSTRACT FROM AUTHOR]