  We report precise mass and density measurements of two extremely hot
sub-Neptune-size planets from the K2 mission using radial velocities, K2
photometry, and adaptive optics imaging. K2-66 harbors a close-in
sub-Neptune-sized (2.49$^{+0.34}_{-0.24} R_\oplus$) planet (K2-66b) with a mass
of 21.3 $\pm$ 3.6 $M_\oplus$. Because the star is evolving up the sub-giant
branch, K2-66b receives a high level of irradiation, roughly twice the main
sequence value. K2-66b may reside within the so-called "photoevaporation
desert", a domain of planet size and incident flux that is almost completely
devoid of planets. Its mass and radius imply that K2-66b has, at most, a meager
envelope fraction (< 5%) and perhaps no envelope at all, making it one of the
largest planets without a significant envelope. K2-106 hosts an
ultra-short-period planet ($P$ = 13.7 hrs) that is one of the hottest
sub-Neptune-size planets discovered to date. Its radius (1.82$^{+0.20}_{-0.14}
R_\oplus$) and mass (9.0 $\pm$ 1.6 $M_\oplus$) are consistent with a rocky
composition, as are all other small ultra-short-period planets with
well-measured masses. K2-106 also hosts a larger, longer-period planet (Rp =
2.77$^{+0.37}_{-0.23} R_\oplus$, $P$ = 13.3 days) with a mass less than 24.4
$M_\oplus$ at 99.7% confidence. K2-66b and K2-106b probe planetary physics in
extreme radiation environments. Their high densities reflect the challenge of
retaining a substantial gas envelope in such extreme environments.
