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 H \beta 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 \dot { \mathscr { M } } \gtrsim 200 , where \dot { \mathscr { M } } = \dot { M } _ { \bullet } / L _ { Edd } c ^ { -2 } , \dot { M } _ { \bullet } is the mass accretion rates , L _ { Edd } is the Eddington luminosity and c is the speed of light . We find that the H \beta 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 ( R _ { { } _ { H \beta } } ) and optical luminosity at 5100Å , R _ { { } _ { H \beta } } - L _ { 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 R _ { { } _ { H \beta } } by the gravitational radius of the black hole , we define a new radius-mass parameter ( Y ) and show that it saturates at a critical accretion rate of \dot { \mathscr { M } } _ { c } = 6 \sim 30 , indicating a transition from thin to slim accretion disk and a saturated luminosity of the slim disks . The parameter Y is a very useful probe for understanding the various types of accretion onto massive black holes . We briefly comment on implications to the general population of super-Eddington AGNs in the universe and applications to cosmology .