  Generic models of regular black holes have separate outer and inner horizons,
both with nonzero surface gravity. It has been shown that a nonzero inner
horizon surface gravity results in exponential instability at the inner horizon
controlled by this parameter. This phenomenon takes the name of "mass inflation
instability", and its presence has put in question the physical viability of
regular black holes as alternatives to their (singular) general relativity
counterparts. In this paper, we show that it is possible to make the inner
horizon surface gravity vanish, while maintaining the separation between
horizons, and a non-zero outer horizon surface gravity. We construct specific
geometries satisfying these requirements, and analyze their behavior under
different kinds of perturbations, showing that the exponential growth
characteristic of mass inflation instability is not present for these
geometries. These "inner-extremal" regular black holes are thereby better
behaved than singular black holes and generic regular black holes, thus
providing a well-motivated alternative of interest for fundamental and
phenomenological studies.
