A definition of biological complexity is proposed grounded in the topology of interconnectivity among a system’s parts. When new parts and their connections become integrated within a network, the cost of their loss increases with the depth of their interdependency resulting in an asymmetry in which gains require toleration, whereas losses increasingly risk disrupting multiple interdependencies. This ‘ratcheting’ is reversible, i.e., apparent reductions in complexity can occur (e.g., parasitism). The limited published data for cell-type diversity, proteome and interactome size, and bulk-flow mechanisms are consistent with the framework’s predictions, motivating additional detailed empirical analyses. The ratchet is distinguished from, yet complementary to, the Zero Force Evolutionary Law, constructive neutral evolution, generative entrenchment, and the phenotype-first tradition, extending their insights to all levels of biological organization. The quantification of complexity thereby becomes an empirical program that asks not whether complexity must increase, but under what conditions history makes interconnectivities difficult to reduce.