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We present new results on the chemical composition of the Galactic ring nebula NGC 6888 surrounding the WN6(h) star WR136. The data are based on deep spectroscopical observations taken with the High Dispersion Spectrograph at the 8.2 m Subaru Telescope. The spectra cover the optical range from 3700 to 7400 Å. The effect of the CNO cycle is well-identified in the abundances of He, N, and O, while elements not involved in the synthesis such as Ar, S, and Fe present values consistent with the solar vicinity and the ambient gas. The major achievement of this work is the first detection of the faint C II λ4267 recombination line in a Wolf-Rayet nebula. This allows us to estimate the C abundance in NGC 6888 and therefore investigate for the first time the trace of the CNO cycle in a ring nebula around a Wolf-Rayet star. Although the detection of the C II line has a low signal-to-noise ratio, the C abundance seems to be higher than the predictions of recent stellar evolution models of massive stars. The Ne abundance also shows a puzzling pattern with an abundance of about 0.5 dex lower than the solar vicinity, which may be related to the action of the NeNa cycle. Attending to the constraints imposed by the dynamical timescale and the He/H and N/O ratios of the nebula, the comparison with stellar evolution models indicates that the initial mass of the stellar progenitor of NGC 6888 is between 25 M_☉ and 40 M_☉.