  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
$\dot{\mathscr{M}}= \dot{M}_{\bullet}/L_{\rm Edd}c^{-2}$, $\dot{M}_{\bullet}$
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 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 ($R_{_{\rm H\beta}}$) and optical luminosity at
5100\AA, $R_{_{\rm H\beta}}-L_{5100}$, 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 $R_{_{\rm H\beta}}$ 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.
