  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 days, respectively. The nature of the third
transiting planet with a 16.6-day 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, 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\%,
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.
