Fibrillar protein aggregates are a hallmark of a range of human disorders, from prion diseases to dementias, but are also encountered in several functional contexts. Yet, the fundamental links between protein assembly mechanisms and their functional or pathological roles have remained elusive. Here, we analyze the aggregation kinetics of a large set of proteins that self-assemble by a nucleated-growth mechanism, from those associated with disease, over those whose aggregates fulfill functional roles in biology, to those that aggregate only under artificial conditions. We find that, essentially, all such systems, regardless of their biological role, are capable of self-replication. However, for aggregates that have evolved to fulfill a structural role, the rate of self-replication is too low to be significant on the biologically relevant time scale. By contrast, all disease-related proteins are able to self-replicate quickly compared to the time scale of the associated disease. Our findings establish the ubiquity of self-replication and point to its potential importance across aggregation-related disorders.