  The detection of a super-Earth and three mini-Neptunes transiting the bright
($V$ = 9.2 mag) star HD 108236 (also known as TOI-1233) was recently reported
on the basis of TESS and ground-based light curves. We perform a first
characterisation of the HD 108236 planetary system through high-precision
CHEOPS photometry and improve the transit ephemerides and system parameters. We
characterise the host star through spectroscopic analysis and derive the radius
with the infrared flux method. We constrain the stellar mass and age by
combining the results obtained from two sets of stellar evolutionary tracks. We
analyse the available TESS light curves and one CHEOPS transit light curve for
each known planet in the system. We find that HD 108236 is a Sun-like star with
$R_{\star}=0.877\pm0.008 R_{\odot}$, $M_{\star}=0.869^{+0.050}_{-0.048}
M_{\odot}$, and an age of $6.7_{-5.1}^{+4.0}$ Gyr. We report the serendipitous
detection of an additional planet, HD 108236 f, in one of the CHEOPS light
curves. For this planet, the combined analysis of the TESS and CHEOPS light
curves leads to a tentative orbital period of about 29.5 days. From the light
curve analysis, we obtain radii of $1.615\pm0.051$, $2.071\pm0.052$,
$2.539_{-0.065}^{+0.062}$, $3.083\pm0.052$, and $2.017_{-0.057}^{+0.052}$
$R_{\oplus}$ for planets HD 108236 b to HD 108236 f, respectively. These values
are in agreement with previous TESS-based estimates, but with an improved
precision of about a factor of two. We perform a stability analysis of the
system, concluding that the planetary orbits most likely have eccentricities
smaller than 0.1. We also employ a planetary atmospheric evolution framework to
constrain the masses of the five planets, concluding that HD 108236 b and HD
108236 c should have an Earth-like density, while the outer planets should host
a low mean molecular weight envelope.
