Long-lived fluxon excitations can be trapped inside a superinductor ring, which is divided into an array of loops by a periodic sequence of Josephson junctions in the quantum regime, thereby allowing fluxons to tunnel between neighboring sites. By tuning the Josephson couplings and, implicitly, the fluxon tunneling probability amplitudes, a wide class of one-dimensional tight-binding lattice models may be implemented and populated with a stable number of fluxons. We illustrate the use of this quantum simulation platform by discussing the Su-Schrieffer-Heeger model in the one-fluxon subspace, which hosts a symmetry-protected topological phase with fractionally charged bound states at the edges. This pair of localized edge states could be used to implement a superconducting qubit increasingly decoupled from decoherence mechanisms.