  We extend the concept of a finite dimensional {\it holomorphic homogeneous
regular} (HHR) domain and some of its properties to the infinite dimensional
setting. In particular, we show that infinite dimensional HHR domains are
domains of holomorphy and determine completely the class of infinite
dimensional bounded symmetric domains which are HHR. We compute the greatest
lower bound of the squeezing function of all HHR bounded symmetric domains,
including the two exceptional domains. We also show that uniformly elliptic
domains in Hilbert spaces are HHR.
