  We present radial velocity follow-up obtained with ESPRESSO of the M-type
star LTT 1445A (TOI-455), for which a transiting planet b with an orbital
period of~5.4 days was detected by TESS. We report the discovery of a second
transiting planet (LTT 1445A c) and a third non-transiting candidate planet
(LTT 1445A d) with orbital periods of 3.12 and 24.30 days, respectively. The
host star is the main component of a triple M-dwarf system at a distance of 6.9
pc. We used 84 ESPRESSO high-resolution spectra to determine accurate masses of
2.3$\pm$0.3 $\mathrm{M}_\oplus$ and 1.0$\pm$0.2 $\mathrm{M}_\oplus$ for planets
b and c and a minimum mass of 2.7$\pm$0.7 $\mathrm{M}_\oplus$ for planet d.
  Based on its radius of 1.43$\pm0.09$ $\mathrm{R}_\oplus$ as derived from the
TESS observations, LTT 1445A b has a lower density than the Earth and may
therefore hold a sizeable atmosphere, which makes it a prime target for the
James Webb Space Telescope. We used a Bayesian inference approach with the
nested sampling algorithm and a set of models to test the robustness of the
retrieved physical values of the system. There is a probability of 85$\%$ that
the transit of planet c is grazing, which results in a retrieved radius with
large uncertainties at 1.60$^{+0.67}_{-0.34}$ $\mathrm{R}_\oplus$. LTT 1445A d
orbits the inner boundary of the habitable zone of its host star and could be a
prime target for the James Webb Space Telescope.
