Context. The TOI-178 system consists of a nearby late K-dwarf transited by six planets in the super-Earth to mini-Neptune regime, with radii ranging from ~1.1 to 2.9 R⊕ and orbital periods between 1.9 and 20.7 days. All planets but the innermost one form a chain of Laplace resonances. Mass estimates derived from a preliminary radial velocity (RV) dataset suggest that the planetary densities do not decrease in a monotonic way with the orbital distance to the star, contrary to what one would expect based on simple formation and evolution models. Aims. To improve the characterisation of this key system and prepare for future studies (in particular with JWST), we performed a detailed photometric study based on 40 new CHEOPS visits, one new TESS sector, and previously published CHEOPS, TESS, and NGTS data. Methods. First we updated the parameters of the host star using the new parallax from Gaia EDR3. We then performed a global analysis of the 100 transits contained in our data to refine the physical and orbital parameters of the six planets and study their transit timing variations (TTVs). We also used our extensive dataset to place constraints on the radii and orbital periods of potential additional transiting planets in the system. Results. Our analysis significantly refines the transit parameters of the six planets, most notably their radii, for which we now obtain relative precisions of ≲3%, with the exception of the smallest planet, b, for which the precision is 5.1%. Combined with the RV mass estimates, the measured TTVs allow us to constrain the eccentricities of planets c to g, which are found to be all below 0.02, as expected from stability requirements. Taken alone, the TTVs also suggest a higher mass for planet d than that estimated from the RVs, which had been found to yield a surprisingly low density for this planet. However, the masses derived from the current TTV dataset are very prior-dependent, and further observations, over a longer temporal baseline, are needed to deepen our understanding of this iconic planetary system.