From high-resolution VLT/UVES spectra (R~43,000), we determine abundances or upper limits for Li, C, N, O, and other important elements, as well as 12C/13C isotopic ratio for three Carbon enhanced metal poor stars. All three stars have -3.30 <= Fe/H <= -2.85 and moderate to high CNO abundances. CS 22958-042 is one of the most carbon-rich CEMP stars known (C/Fe = +3.2), while CS 29528-041 (one of the few N-enhanced metal-poor stars known) is one of the most nitrogen rich (N/Fe = +3.0). Oxygen is very high in CS 31080-095 (O/Fe = +2.35) and in CS 22958-042 (O/Fe = +1.35). All three stars exhibit Sr/Fe < 0; Ba is not detected in CS 22958-042 (Ba/Fe < -0.53), but it is moderately enhanced (Ba/Fe ~ 1) in the other two stars. CS 22958-042 displays one of the largest sodium overabundances yet found in CEMP stars (Na/Fe = +2.8). CS 22958-042 has 12C/13C = 9, similar to most other CEMP stars without enhanced neutron-capture elements, while 12C/13C <= 40 in CS 31080-095. CS 31080-095 and CS 29528-041 have A(Li) ~ 1.7, below the Spite Plateau, while Li is not detected in CS 22958-042. CS 22958-042 is a CEMP-no star, but the other two stars are in no known class of CEMP star and thus either constitute a new class or are a link between the CEMP-no and CEMP-s classes, adding complexity to the abundance patterns for CEMP stars. We interpret the abundance patterns in our stars to imply that current models for the presumed AGB binary progenitors lack an extra-mixing process, similar to those apparently operating in RGB stars.