The cold gas reservoir surrounding present day massive galaxies like MW is generally insufficient to explain their current star formation rate or their baryonic mass assembly history. Therefore, additional mass accretion from cold flow and/or gas-rich neighbors is necessary in order to account for the observed properties. This problem becomes more severe in the earlier epochs (z~1 to 3), when the average star formation rate is observed to be higher, and hence the gas mass fraction is expected to be larger. Intriguingly, extremely gas-rich galaxies are being found as we have become able to probe the gas content of more distant galaxies. In this talk, I will present the molecular gas properties of a sample of local (z~0.05) analogs with a large cold gas reservoir, dubbed “HI monsters”, and discuss their implications for the galaxy growth.