Fibrillar protein aggregates are a hallmark of the pathology of a range of human disorders, from prion diseases to dementias. Yet, the same aggregated structures that are formed in disease are also encountered in several functional contexts. The fundamental properties that determine whether these protein assembly processes are functional or, by contrast, pathological, have remained elusive. Here, we address this question by analysing the aggregation kinetics of a large set of self-assembling proteins, from those associated with disease, over those whose aggregates fulfil functional roles in biology, to those that aggregate only under artificial conditions. Remarkably, we find that essentially all systems that assemble by a nucleated-growth mechanism are capable of significant self-replication on experimentally accessible timescales. However, comparing the intrinsic timescales of self-replication with the timescales over which the corresponding aggregates form in a biological context yields a clear distinction; for aggregates which have evolved to fulfil a structural role, the rate of self-replication is too low to be significant on the biologically relevant timescale. By contrast, all analysed proteins that aggregate in the context of disease are able to self-replicate quickly compared to the timescale of the associated disease. Our findings establish the ability to self-replicate as both a ubiquitous property of protein aggregates and one that has the potential to be a key process across aggregation-related disorders.