We report the discovery of a third planet transiting the star TOI-1260, previously known to host two transiting sub-Neptune planets with orbital periods of 3.127 and 7.493 d, respectively. The nature of the third transiting planet with a 16.6-d orbit is supported by ground-based follow-up observations, including time-series photometry, high-angular resolution images, spectroscopy, and archival imagery. Precise photometric monitoring with CHEOPS allows to improve the constraints on the parameters of the system, improving our knowledge on their composition. The improved radii of TOI-1260b and TOI-1260c are $2.36 \pm 0.06 \rm R_{\oplus }$, $2.82 \pm 0.08 \rm R_{\oplus }$, respectively while the newly discovered third planet has a radius of $3.09 \pm 0.09 \rm R_{\oplus }$. The radius uncertainties are in the range of 3 per cent, allowing a precise interpretation of the interior structure of the three planets. Our planet interior composition model suggests that all three planets in the TOI-1260 system contains some fraction of gas. The innermost planet TOI-1260b has most likely lost all of its primordial hydrogen-dominated envelope. Planets c and d were also likely to have experienced significant loss of atmospheric through escape, but to a lesser extent compared to planet b.