Multistage spin correlations in the $s$ = 1/2 stuffed hyper-star lattice Li$_{2}$Cu$_{2}$(MoO$_{4}$)$_{3}$
Star lattice, which can be visualized as a honeycomb network with each vertex replaced by a triangle, provides a rare platform for realizing exotic quantum states such as quantum spin liquids and disorder-driven random-singlet (RS) states. Herein, we investigate the ground-state properties of the three-dimensional (3D) stuffed hyper-star lattice Li$_2$Cu$_2$(MoO$_4$)$_3$, which exhibits a crossover from short-range spin correlations to a disorder-driven RS-like state below $T^{*}\sim$15.8 K. Thermodynamic and microscopic measurements capture this crossover through a change in the power-law behavior of various observables, from $\sim T^{0.25}$ for $T > T^{*}$ to $\sim T^{-0.50}$ for $T < T^{*}$. Upon further cooling, a quasi-frozen state emerges near $T_{\rm f} = 0.32$ K, likely associated with weakly coupled spin chains within the hyper-star spin network. Our results underscore the crucial role of orphan spins and weak residual interactions in stabilizing a disorder-driven quantum-disordered state in 3D.