Context. Historically, galaxy bulges were thought to be single-component objects at the center of galaxies. However, this picture is now questioned since different bulge types with different formation paths, namely classical and pseudobulges, have been found coexisting within the same galaxy. Aims: We study the incidence and 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. Methods: 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 characterized. Results: We find only three galaxies hosting a single-component bulge (two pseudobulges and one classical bulge). Thus, we demonstrate the high incidence of composite bulges (70%) in barred galaxies. We find evidence of composite bulges coming in two main types based on their formation: secular-built and merger- and secular-built. We denote as secular-built those composite bulges that are made up of structures associated with secular processes, such as pseudobulges, 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 notice the importance of detecting kinematic structures such as σ-drops to identify composite bulges. A high percentage (~80%) of galaxies were found to host σ-drops or σ-plateaus in our sample, revealing their high incidence in barred galaxies. Conclusions: The high frequency of composite bulges in barred galaxies points toward a complex formation and evolutionary scenario. Moreover, the evidence of 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.