  Stellar population models produce radiation fields that ionize oxygen up to
O+2, defining the limit of standard HII region models (<54.9 eV). Yet, some
extreme emission line galaxies, or EELGs, have surprisingly strong emission
originating from much higher ionization potentials. We present UV-HST/COS and
optical-LBT/MODS spectra of two nearby EELGs that have very-high-ionization
emission lines (e.g., HeII 1640,4686 CIV 1548,1550, [FeV] 4227, [ArIV]
4711,4740). We define a 4-zone ionization model that is augmented by a
very-high-ionization zone, as characterized by He+2 (>54.4 eV).
  The 4-zone model has little to no effect on the measured total nebular
abundances, but does change the interpretation of other EELG properties: we
measure steeper central ionization gradients, higher volume-averaged ionization
parameters, and higher central T_e, n_e, and logU values. Traditional 3-zone
estimates of the ionization parameter can under-estimate the average log U by
up to 0.5 dex. Additionally, we find a model-independent dichotomy in the
abundance patterns, where the alpha/H-abundances are consistent but N/H, C/H,
and Fe/H are relatively deficient, suggesting these EELGs are alpha/Fe-enriched
by >3 times. However, there still is a high-energy ionizing photon production
problem (HEIP^3). Even for such alpha/Fe-enrichment and very-high log Us,
photoionization models cannot reproduce the very-high-ionization emission lines
observed in EELGs.
