We present a detailed analysis of three extremely strong, intervening damped Lyman-α systems (ESDLAs, with log N(H i) ≥ 21.7) observed towards quasars with the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope. We measure overall metallicities of [Zn/H] ∼ −1.2, −1.3, and −0.7 at, respectively, z abs = 2.34 towards SDSS J214043.02−032139.2 (log N(H i) = 22.4 ± 0.1), z abs = 3.35 towards SDSS J145646.48+160939.3 (log N(H i) = 21.7 ± 0.1), and z abs = 2.25 towards SDSS J015445.22+193515.8 (log N(H i) = 21.75 ± 0.15). Iron depletion of about a factor 15 compared to volatile elements is seen in the DLA towards J2140−0321, while the other two show deletion that is typical of known DLAs. We detect H 2 towards J2140−0321 (log N(H 2 ) = 20.13 ± 0.07) and J1456+1609 (log N(H 2 ) = 17.10 ± 0.09) and argue for a tentative detection towards J0154+1935. Absorption from the excited fine-structure levels of O i, C i, and Si ii are detected in the system towards J2140−0321, which has the largest H i column density detected so far in an intervening DLA. This is the first detection of O i fine-structure lines in a QSO-DLA, which also provides us with 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: H 2 and C i most likely stem from a dense (log n H ∼ 2.5−3) and cold (80 K) phase and from a warm (250 K) phase. They contain a fraction of the total H i, while a warmer (T > 1000 K) phase probably contributes significantly to the high excitation of O i fine-structure levels. The observed C i/H 2 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. near-infrared emission lines in the surroundings of the gas would enable a detailed physical modelling of the system.