Post-asymptotic giant branch (post-AGB) stars are known to be chemically diverse. In this paper we present the first observational evidence of a star that has failed the third dredge-up (TDU). J005252.87-722842.9 is a A-type ($T_{rm eff}$ = 8250 $pm$ 250K) luminous (8200 $pm$ 700 $rm L_{odot}$), metal-poor ($textrm{[Fe/H]}$ = $- 1.18 pm$ 0.10), low-mass (M$_{rm initial}$ $approx$ 1.5 $-$ 2.0 $rm M_{odot}$) post-AGB star in the Small Magellanic Cloud. Through a systematic abundance study, using high-resolution optical spectra from UVES, we found that this likely post-AGB object shows an intriguing photospheric composition with no confirmed carbon-enhancement (upper limit of [C/Fe] $<$ 0.50) nor enrichment of $s$-process elements. We derived an oxygen abundance of [O/Fe] = 0.29 $pm$ 0.1. For Fe and O, we took into account the effects of non-local thermodynamic equilibrium (NLTE). We could not derive an upper limit for the nitrogen abundance as there are no useful nitrogen lines within our spectral coverage. The chemical pattern displayed by this object has not been observed in single or binary post-AGBs. Based on its derived stellar parameters and inferred evolutionary state, single star nucleosynthesis models predict that this star should have undergone TDU episodes while on the AGB and be carbon-enriched. However, our observations are in contrast with these predictions. We identify two possible Galactic analogues which are likely to be post-AGB stars, but the lack of accurate distances (hence luminosities) to these objects does not allow us to confirm their post-AGB status. If they have low luminosities then they are likely to be dusty post-RGB stars. The discovery of J005252.87-722842.9 reveals a new stellar evolutionary channel whereby a star evolves without any third dredge-up episodes.
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