We demonstrate that Li+ hopping conduction, which cannot be explained by conventional models i.e., Onsager's theory and Stokes' law, emerges in highly concentrated liquid electrolytes composed of LiBF4 and sulfolane (SL). Self-diffusion coefficients of Li+ (D-Li), BF4- (D-BF4), and SL (D-SL) were measured with pulsed-field gradient NMR. In the concentrated electrolytes with molar ratios of SL/LiBF4 <= 3, the ratios D-SL/D-Li and D-BF4/D-Li become lower than 1, suggesting faster diffusion of Li+ than SL and BF4-, and thus the evolution of Li+ hopping conduction. X-ray crystallographic analysis of the LiBF4/SL (1:1) solvate revealed that the two oxygen atoms of the sulfone group are involved in the bridging coordination of two different Li+ ions. In addition, the BF4- anion also participates in the bridging coordination of Li+. The Raman spectra of the highly concentrated LiBF4-SL solution suggested that Li+ ions are bridged by SL and BF4- even in the liquid state. Moreover, detailed investigation along with molecular dynamics simulations suggests that Li+ exchanges ligands (SL and BF4-) dynamically in the highly concentrated electrolytes, and Li+ hops from one coordination site to another. The spatial proximity of coordination sites, along with the possible domain structure, is assumed to enable Li+ hopping conduction. Finally, we demonstrate that Li+ hopping suppresses concentration polarization in Li batteries, leading to increased limiting current density and improved rate capability compared to the conventional concentration electrolyte. Identification and rationalization of Li+ ion hopping in concentrated SL electrolytes is expected to trigger a new paradigm of understanding for such unconventional electrolyte systems.