We present a detailed analysis of three extremely strong intervening DLAs (log N(HI)>=21.7) observed towards quasars with VLT/UVES. We measure overall metallicities of [Zn/H]~-1.2, -1.3 and -0.7 at respectively zabs=2.34 towards SDSS J2140-0321 (log N(HI) = 22.4+/-0.1), zabs=3.35 towards SDSS J1456+1609 (log N(HI) = 21.7+/-0.1) and zabs=2.25 towards SDSS J0154+1935 (log N(HI) = 21.75+/-0.15). We detect H2 towards J2140-0321 (log N(H2) = 20.13+/-0.07) and J1456+1609 (log N(H2) = 17.10+/-0.09) and argue for a tentative detection towards J0154+1935. Absorption from the excited fine-structure levels of OI, CI and SiII are detected in the system towards J2140-0321, that has the largest HI column density detected so far in an intervening DLA. This is the first detection of OI fine-structure lines in a QSO-DLA, that also provides us a rare possibility to study the chemical abundances of less abundant atoms like Co and Ge. Simple single phase photo-ionisation models fail to reproduce all the observed quantities. Instead, we suggest that the cloud has a stratified structure: H2 and CI likely originate from both a dense (log nH~2.5-3) cold (80K) and warm (250K) phase containing a fraction of the total HI while a warmer (T>1000 K) phase probably contributes significantly to the high excitation of OI fine-structure levels. The observed CI/H2 column density ratio is surprisingly low compared to model predictions and we do not detect CO molecules: this suggests a possible underabundance of C by 0.7 dex compared to other alpha elements. The absorber could be a photo-dissociation region close to a bright star (or a star cluster) where higher temperature occurs in the illuminated region. Direct detection of on-going star formation through e.g. NIR emission lines in the surrounding of the gas would enable a detailed physical modelling of the system.
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