The full version of the equation-of-motion coupled-cluster (EOMCC) method with all singles and doubles, and a selected set of triples defined through active orbitals (EOMCCSDt) has been implemented and tested using the H-8, H2O, N-2, C-2, and CH+ systems. It is demonstrated that the full EOMCCSDt method provides the results of the full EOMCCSDT (EOMCC singles, doubles, and triples) quality at the fraction of the computer effort associated with the EOMCCSDT calculations. This includes excited states that are dominated by doubles and states that have large triexcited components. The excellent performance of the EOMCCSDt approach is observed even when the ground electronic state has a quasidegenerate character, which means that we can apply the EOMCCSDt formalism to excited states that cannot be adequately described by the perturbative triples models. The EOMCCSDt method is equivalent to the EOMCCSDT approach if all orbitals used in the EOMCCSDt calculations are active.