Context. Thick discs can give invaluable information on the formation and evolution history of galaxies as most, if not all, disc galaxies have a thin (classical) disc and a thick disc. Aims. We study the structure of thick discs in extraordinary depth by reaching a surface brightness limit of μ_rdeep ∼ 28.5−29 mag arcsec⁻² with combined g, r, i band images from the IAC Stripe 82 Legacy Project. Methods. We present the characterisation of the thick discs in a sample of five edge-on galaxies. A study of the radial and vertical surface brightness profiles is presented by comparing our data with point spread function (PSF) deconvolved models. Our method begins with an analysis of the background and masking processes. Then we consider the effects of the PSF through galaxy modelling. The galaxy disc components are fitted considering that the thin and thick discs are two stellar fluids that are gravitationally coupled in hydrostatic equilibrium. Results. We find that effects due to the PSF are significant when low surface brightness is reached, especially in the vertical profiles, but it can be accounted for by careful modelling. The galaxy outskirts are strongly affected by the faint wings of the PSF, mainly by PSF-redistributed light from the thin disc. This is a central problem for ultra-deep imaging. The thick-disc component is required to reach satisfactory fit results in the more complex galaxies in our sample, although it is not required for all galaxies. When the PSF is ignored, the brightness of these structures may be overestimated by up to a factor of ∼4. Conclusions. In general, our results are in good agreement with those of previous works, although we reach deeper surface brightness levels, so that the PSF effects are stronger. We obtain scale heights and mass ratios of thin and thick discs (z_t, z_T, and M_T/M_t), which provide excellent agreement with previous studies. Our small initial sample provides evidence for aspects of a wide variety of formation theories for the thick discs in disc galaxies.