  (Abridged) We study the incidence, as well as the nature, of composite bulges
in a sample of 10 face-on barred galaxies to constrain the formation and
evolutionary processes of the central regions of disk galaxies. We analyze the
morphological, photometric, and kinematic properties of each bulge. Then, by
using a case-by-case analysis we identify composite bulges and classify every
component into a classical or pseudobulge. In addition, bar-related boxy/peanut
(B/P) structures were also identified and characterised. We find only three
galaxies hosting a single-component bulge (two pseudobulges and one classical
bulge). We find evidence of composite bulges coming in two main types based on
their formation: secular-built and merger- and secular-built. We call
secular-built to composite bulges made of entirely by structures associated
with secular processes such as pseudo bulges, central disks, or B/P bulges. We
find four composite bulges of this kind in our sample. On the other hand,
merger- and secular-built bulges are those where structures with different
formation paths coexist within the same galaxy, i.e., a classical bulge
coexisting with a secular-built structure (pseudobulge, central disk, or B/P).
Three bulges of this kind were found in the sample. We remark on the importance
of detecting kinematic structures such as sigma-drops to identify composite
bulges. A large fraction (80%) of galaxies were found to host sigma-drops or
sigma-plateaus in our sample revealing their high incidence in barred galaxies.
The high frequency of composite bulges in barred galaxies points towards a
complex formation and evolutionary scenario. Moreover, the evidence for
coexisting merger- and secular-built bulges reinforce this idea. We discuss how
the presence of different bulge types, with different formation histories and
timescales, can constrain current models of bulge formation.
