  While all media can exhibit first-order conductivity describing current
linearly proportional to electric field, $E$, the second-order conductivity,
$\sigma^{(2)}$ , relating current to $E^2$, is nonzero only when inversion
symmetry is broken. Second order nonlinear optical responses are powerful tools
in basic research, as probes of symmetry breaking, and in optical technology as
the basis for generating currents from far-infrared to X-ray wavelengths. The
recent surge of interest in Weyl semimetals with acentric crystal structures
has led to the discovery of a host of $\sigma^{(2)}$ -related phenomena in this
class of materials, such as polarization-selective conversion of light to dc
current (photogalvanic effects) and the observation of giant second-harmonic
generation (SHG) efficiency in TaAs at photon energy 1.5 eV. Here, we present
measurements of the SHG spectrum of TaAs revealing that the response at 1.5 eV
corresponds to the high-energy tail of a resonance at 0.7 eV, at which point
the second harmonic conductivity is approximately 200 times larger than seen in
the standard candle nonlinear crystal, GaAs. This remarkably large SHG response
provokes the question of ultimate limits on $\sigma^{(2)}$ , which we address
by a new theorem relating frequency-integrated nonlinear response functions to
the third cumulant (or "skewness") of the polarization distribution function in
the ground state. This theorem provides considerable insight into the factors
that lead to the largest possible second-order nonlinear response, specifically
showing that the spectral weight is unbounded and potentially divergent when
the possibility of next-neighbor hopping is included.
