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ABSTRACT We have completed two years of photometric and spectroscopic monitoring of a large number of active galactic nuclei (AGNs) with very high accretion rates. In this paper, we report on the result of the second phase of the campaign, during 2013-2014, and the measurements of five new Hβ time lags out of eight monitored AGNs. All five objects were identified as super-Eddington accreting massive black holes (SEAMBHs). The highest measured accretion rates for the objects in this campaign are , where , is the mass accretion rates, is the Eddington luminosity and c is the speed of light. We find that the Hβ time lags in SEAMBHs are significantly shorter than those measured in sub-Eddington AGNs, and the deviations increase with increasing accretion rates. Thus, the relationship between broad-line region size ( ) and optical luminosity at 5100 , , requires accretion rate as an additional parameter. We propose that much of the effect may be due to the strong anisotropy of the emitted slim-disk radiation. Scaling by the gravitational radius of the black hole (BH), we define a new radius-mass parameter ( ) and show that it saturates at a critical accretion rate of , indicating a transition from thin to slim accretion disk and a saturated luminosity of the slim disks. The parameter is a very useful probe for understanding the various types of accretion onto massive BHs. We briefly comment on implications to the general population of super-Eddington AGNs in the universe and applications to cosmology.