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_(-0.24)^(+0.34)R_⊕) planet (K2-66b) with a mass of 21.3 ± 3.6 M_⊕. Because the star is evolving up the subgiant 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 hr) that is one of the hottest sub-Neptune-size planets discovered to date. Its radius (1.82_(-0.14)^(+0.20) R_⊕) and mass (9.0 ± 1.6 M_⊕) 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 (R_p = 2.77_(-0.23^(+0.37)R_⊕, P = 13.3 days) with a mass less than 24.4 M_⊕ 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
