  We calculate the trispectrum T_\zeta of the primordial curvature perturbation
\zeta, generated during a {\it slow-roll} inflationary epoch by considering a
two-field quadratic model of inflation with {\it canonical} kinetic terms. We
consider loop contributions as well as tree level terms, and show that it is
possible to attain very high, {\it including observable}, values for the level
of non-gaussianity \tau_{NL} if T_\zeta is dominated by the one-loop
contribution. Special attention is paid to the claim in JCAP {\bf 0902}, 017
(2009) [arXiv:0812.0807 [astro-ph]] that, in the model studied in this paper
and for the specific inflationary trajectory we choose, the quantum
fluctuations of the fields overwhelm the classical evolution. We argue that
such a claim actually does not apply to our model, although more research is
needed in order to understand the role of quantum diffusion. We also consider
the probability that an observer in an ensemble of realizations of the density
field sees a non-gaussian distribution. In that respect, we show that the
probability associated to the chosen inflationary trajectory is non-negligible.
Finally, the levels of non-gaussianity f_{NL} and \tau_{NL} in the bispectrum
B_\zeta and trispectrum T_\zeta of \zeta, respectively, are also studied for
the case in which \zeta is not generated during inflation.

Astrophysics;General Relativity and Quantum Cosmology;High Energy Physics - Phenomenology