High densities exceeding the Greenwald limit by a factor of 1.7 have been obtained in L-mode discharges with high internal inductances of ℓi as high as 2.8 in JT-60U. The internal inductance is controlled by ramping down the plasma current. In addition to the extension of the operational regime limited by disruptions, confinement performance remains as good as an H89PL factor of 1.6 even above the Greenwald limit. While an earlier high ℓi study has indicated core confinement improvement due to enhancement of the poloidal field, the additional improvement of the tolerance against the high density turned out to be correlated with high edge temperature. The normalized density when the detachment occurs, characterized by a decrease in the Dα signal at the divertor, is even higher in the case with no disruption than in the case with a disruption. These comparisons have indicated that the improvement in thermal and particle transport does exist in the periphery and in the edge in high ℓi plasmas, and the shift of the density limit towards higher densities is observed coincidently. Although the high ℓi discharge studied here lies outside the usual parameter space for steady-state operation of a tokamak, demonstration of a stable discharge with good confinement beyond the Greenwald limit suggests that the magnetic shear at the edge is one key parameter to uncover the physical elements of the operational density limit.