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 ± 0.3 M ⊕ and 1.0 ± 0.2 M ⊕ for planets b and c and a minimum mass of 2.7 ± 0.7 M ⊕ for planet d. Based on its radius of 1.43 ± 0.09 R ⊕ 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 (JWST). 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 −0.34 +0.67 R ⊕ . LTT 1445A d orbits the inner boundary of the habitable zone of its host star and could be a prime target for the JWST.