  The renormalon singularities are a known source of the divergent behavior of
asymptotic perturbative series from field theoretical models. These
singularities live in the Borel plane and are responsible for ambiguities in
the physical plane. We propose that field theories can have renormalons beyond
the usual first Borel plane. We show an example with a scalar field theory
where, considering a chain of cat's eyes diagrams, the model develops a
Gevrey-3 asymptotic series.
