The central engine of fast radio bursts (FRB) is not yet understood. Due to the interaction between the pulse and the intervening plasma, the dispersion sweep of FRBs provides a unique probe of its environment and the ionized baryon content of the intergalactic medium. Active repeaters has been shown to be associated with persistent radio source (PRS), and dense, energetic, magnetized plasmas. Here we report the discovery and localization of a new, extremely active repeater, FRB 190520, which is co-located with a compact PRS and identified with a dwarf host galaxy of high star formation at a redshift z=0.241. The estimated host galaxy contribution DMhost≈912+69−108pc cm−3 is nearly an order of magnitude higher than the average of FRB host galaxies and much larger than the contribution from the intergalactic medium, suggesting caution in inferring redshifts for FRBs without accurate host galaxy identifications. This represents the second source after FRB 121102 with confirmed association between FRB and compact PRS. The dense, complex host galaxy environment and the association with PRS may point to a distinctive origin or an earlier evolutionary stage for highly active repeating FRBs.