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.