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We have completed two years of photometric and spectroscopic monitoring of alarge 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 ofeight monitored AGNs. All five objects were identified as super-Eddingtonaccreting massive black holes (SEAMBHs). The highest measured accretion ratesfor the objects in this campaign are $\dot{\mathscr{M}}\gtrsim 200$, where$\dot{\mathscr{M}}= \dot{M}_{\bullet}/L_{\rm Edd}c^{-2}$, $\dot{M}_{\bullet}$is the mass accretion rates, $L_{\rm Edd}$ is the Eddington luminosity and $c$is the speed of light. We find that the H$\beta$ time lags in SEAMBHs aresignificantly shorter than those measured in sub-Eddington AGNs, and thedeviations increase with increasing accretion rates. Thus, the relationshipbetween broad-line region size ($R_{_{\rm H\beta}}$) and optical luminosity at5100\AA, $R_{_{\rm H\beta}}-L_{5100}$, requires accretion rate as an additionalparameter. We propose that much of the effect may be due to the stronganisotropy of the emitted slim-disk radiation. Scaling $R_{_{\rm H\beta}}$ bythe gravitational radius of the black hole, we define a new radius-massparameter ($Y$) and show that it saturates at a critical accretion rate of$\dot{\mathscr{M}}_c=6\sim 30$, indicating a transition from thin to slimaccretion disk and a saturated luminosity of the slim disks. The parameter $Y$is a very useful probe for understanding the various types of accretion ontomassive black holes. We briefly comment on implications to the generalpopulation of super-Eddington AGNs in the universe and applications tocosmology.

Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. IV. H$\beta$ Time Lags and Implications for Super-Eddington Accretion