  Kepler has found hundreds of Neptune-size (2-6 R_Earth) planet candidates
within 0.5 AU of their stars. The nature of the vast majority of these planets
is not known because their masses have not been measured. Using theoretical
models of planet formation, evolution and structure, we explore the range of
minimum plausible masses for low-density exo-Neptunes. We focus on highly
irradiated planets with T_eq>=500K. We consider two separate formation pathways
for low-mass planets with voluminous atmospheres of light gases: core nucleated
accretion and outgassing of hydrogen from dissociated ices. We show that
Neptune-size planets at T_eq=500K with masses as small as a few times that of
Earth can plausibly be formed core nucleated accretion coupled with subsequent
inward migration. We also derive a limiting low-density mass-radius relation
for rocky planets with outgassed hydrogen envelopes but no surface water. Rocky
planets with outgassed hydrogen envelopes typically have computed radii well
below 3 R_Earth. For both planets with H/He envelopes from core nucleated
accretion and planets with outgassed hydrogen envelopes, we employ planet
interior models to map the range of planet mass--envelope mass--equilibrium
temperature parameter space that is consistent with Neptune-size planet radii.
Atmospheric mass loss mediates which corners of this parameter space are
populated by actual planets and ultimately governs the minimum plausible mass
at a specified transit radius. We find that Kepler's 2-6 R_Earth planet
candidates at T_eq=500--1000K could potentially have masses less than ~4
M_Earth. Although our quantitative results depend on several assumptions, our
qualitative finding that warm Neptune-size planets can have masses
substantially smaller than those given by interpolating the masses and radii of
planets within our Solar System is robust.
