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Two unusual Li-rich K giants, HDE 233517 and HD 9746, have been studied. Optical spectroscopy and photometry have been obtained to determine the fundamental parameters of HDE 233517, a single K2 III with an extremely large infrared excess. The spectra yield Teff=4475 K, logg=2.25, [Fe/H]=-0.37, vsini=17.6 km s-1, and a non-LTE logɛ(7Li)=4.22. Photometric observations reveal low-amplitude light variability with a period of 47.9 days. Combined with other parameters, this results in a minimum radius of 16.7 Rsolar and minimum distance of 617 pc. Comparison of spectra obtained in 1994 and 1996 show profile variations in Hα and the Na D lines indicative of changing mass loss. Optical spectra of HD 9746, a chromospherically active giant, were analyzed. The Teff=4400 K and revised Hipparcos-based gravity of logg=2.30 lead to a non-LTE logɛ(7Li)=3.75. The Li abundances in both stars are supermeteoritic. By the inclusion and exclusion of 6Li in the syntheses, we show that consistent 7Li abundances are obtained only when 6Li is absent in the synthetic fit. This provides evidence for fresh 7Li production and excludes both preservation of primordial Li and planetary accretion as viable scenarios for the formation of Li-rich giants. Both stars lie in close proximity to the red giant luminosity bump supporting the hypothesis that 7Li production is caused by the same mixing mechanism that later results in CN processing and lowers the 12C/13C ratio to nonstandard values.