  We present an extension of the random--phase approximation (RPA) where the
RPA phonons are used as building blocks to construct the excited states. In our
model, that we call double RPA (DRPA), we include up to two RPA phonons. This
is an approximate and simplified way, with respect to the full second
random--phase approximation (SRPA), to extend the RPA by including two
particle--two hole configurations. Some limitations of the standard SRPA model,
related to the violation of the stability condition, are not encountered in the
DRPA. We also verify in this work that the energy--weighted sum rules are
satisfied. The DRPA is applied to low--energy modes and giant resonances in the
nucleus $^{16}$O. We show that the model (i) produces a global downwards shift
of the energies with respect to the RPA spectra; (ii) provides a shift that is
however strongly reduced compared to that generated by the standard SRPA. This
model represents an alternative way of correcting for the SRPA anomalous energy
shift, compared to a recently developed extension of the SRPA, where a
subtraction procedure is applied. The DRPA provides results in good agreeement
with the experimental energies, with the exception of those low--lying states
that have a dominant two particle--two hole nature. For describing such states,
higher--order calculations are needed.
