In this paper we consider the electrically neutral annulenes CMHM (M=6,10,14,18) and their cations CMHM4+ (M=10,14,18), having the cyclic geometry corresponding to the C-M point group, and described by the electronic Hamiltonian of the Pariser-Parr-Pople (PPP) model (in which the strength of electronic correlations is controlled by the magnitude of the resonance integral beta). We study the ground-state electronic-correlation effects in these systems by means of the single-reference coupled-cluster (CC) theory employing the restricted Hartree-Fock wave function as the reference. It is known that the basic CC technique-the coupled cluster singles and doubles (CCSD) method (for the annulenes equivalent to the CCD method)-breaks down in the strongly correlated regime of annulenes. In this paper we analyze the performance of the standard extensions of the CCD method, taking into account the connected triple (T) and/or quadruple (Q) excitations in the CC operator T: the CCDT, CCDQ, and CCDTQ methods. For comparison we use some modifications of the CCD method that emulate the coupling between the D and Q excitations: the ACP and ACPQ methods. Whenever available, the full configuration-interaction results serve as the reference. We study not only the ground-state correlation energies, but also the quality of the CC amplitudes corresponding to the double excitations, as well as the magnitude of quasidegeneracy effects. Our results provide evidence that not only the CCD, but also the CCDT, CCDQ, and CCDTQ methods break down when the correlation effects become sufficiently strong. This indicates a failure of the standard CC theory (in which the CCD method is gradually augmented by taking into account the T, Q etc. excitations) in the strongly correlated regime of the PPP annulene model.