Context. One of the goals of the Echelle Spectrograph for Rocky Exoplanets and Stable Spectroscopic Observations (ESPRESSO) Guaranteed Time Observations (GTO) consortium is the precise characterisation of a selected sample of planetary systems discovered by TESS. One such target is the K0V star HIP 29442 (TOI-469), already known to host a validated sub-Neptune companion TOI-469.01, which we followed-up with ESPRESSO. Aims. We aim to verify the planetary nature of TOI-469.01 by obtaining precise mass, radius, and ephemeris, and constraining its bulk physical structure and composition. Methods. Following a Bayesian approach, we modelled radial velocity and photometric time series to measure the dynamical mass, radius, and ephemeris, and to characterise the internal structure and composition of TOI-469.01. Results. We confirmed the planetary nature of TOI-469.01 (now renamed HIP 29442 b), and thanks to the ESPRESSO radial velocities we discovered two additional close-in companions. Through an in-depth analysis of the TESS light curve, we could also detect their low signal-to-noise transit signals. We characterised the additional companions, and conclude that HIP 29442 is a compact multi-planet system. The three planets have orbital periods Porb,b = 13.63083 ± 0.00003, Porb,c = 3.53796 ± 0.00003, and Porb,d = 6.42975-0.00010+0.00009 days, and we measured their masses with high precision: mp,b = 9.6 ± 0.8 M⊕, mp,c = 4.5 ± 0.3 M⊕, and mp,d = 5.1 ± 0.4 M⊕. We measured radii and bulk densities of all the planets (the 3σ confidence intervals are shown in parentheses): Rp,b = 3.48-0.08(-0.28)+0.07(+0.19) R⊕ and ρp,b = 1.3 ± 0.2(0.3)g cm-3; Rp,c = 1.58-0.11(-0.34)+0.10(+0.30) R⊕ and ρp,c = 6.3-1.3(-2.7)+1.7(+6.0)g cm-3; Rp,d = 1.37 ± 0.11(-0.43)(+0.32) R⊕ and ρp,d = 11.0-2.4(-6.3)+3.4(+21.0)g cm-3. Due to noisy light curves, we used the more conservative 3σ confidence intervals for the radii as input to the interior structure modelling. We find that HIP 29442 b appears as a typical sub-Neptune, likely surrounded by a gas layer of pure H-He with amass of 0.27-0.17+0.24 M⊕ and a thickness of 1.4 ± 0.5 R⊕. For the innermost companions HIP 29442 c and HIP 29442 d, the model supports an Earth-like composition. Conclusions. The compact multi-planet system orbiting HIP 29442 offers the opportunity to study simultaneously planets straddling the gap in the observed radius distribution of close-in small-size exoplanets. High-precision photometric follow-up is required to obtain more accurate and precise radius measurements, especially for planets c and d. This, together with our determined high-precision masses, will provide the accurate and precise bulk structure of the planets, and enable an accurate investigation of the system's evolution.