The fractalized Hofstadter butterfly energy spectrum predicted for magnetically confined fermions diffracted by a crystal lattice has remained beyond the reach of laboratory-accessible magnetic fields. We find the geometrically frustrated spin system SrCu ₂ (BO ₃ ) ₂ to provide a sterling demonstration of a system in which bosons confined by a magnetic and lattice potential mimic the behavior of fermions in the extreme quantum limit, giving rise to a sequence of plateaus at all magnetization m _z /m _sat = 1/ q ratios 9 ≥ q ≥ 2 and p / q = 2/9 ( m _sat is the saturation magnetization) in magnetic fields up to 85 T and temperatures down to 29 mK, within the sequence of previously identified plateaus at 1/8, 1/4, and 1/3 of the saturated magnetization. We identify this hierarchy of plateaus as a consequence of confined bosons in SrCu ₂ (BO ₃ ) ₂ mimicking the high magnetic field fractalization predicted by the Hofstadter butterfly for fermionic systems. Such an experimental realization of the Hofstadter problem for interacting fermions has not been previously achieved in real materials, given the unachievably high magnetic flux densities or large lattice periods required. By a theoretical treatment that includes short-range repulsion in the Hofstadter treatment, stripe-like spin density-modulated phases are revealed in SrCu ₂ (BO ₃ ) ₂ as emergent from a fluidic fractal spectrum.