Composite systems formed by coupling topological units are emerging as a powerful platform for engineering wave phenomena. However, general principles governing how coupling modifies the topology of such composites remain largely unexplored. Here, we develop an analytically tractable framework to understand and control the topology of composite systems using coupled Su–Schrieffer–Heeger chains as a model. Focusing on an acoustic realization, we discover that the topology of the composite is not uniquely determined by the topological phases of its constituents. Instead, the inter-chain coupling can act as an independent and decisive control parameter that governs the topological character of the composite, regardless of the topological characters of the individual constituents. The composite topological character can also be changed by variations of the intra-chain parameters that preserve the topological characters of the constituent chains. By deriving closed-form expressions for the dispersion relations, corresponding eigenvectors, and Berry phases, we uncover distinct classes of bands whose topologies can either be governed only by intra-chain parameters or can be tuned solely through coupling. Remarkably, this enables regimes in which two trivial (non-trivial) constituents combine to form a non-trivial (trivial) composite. This work establishes flexible principles to achieve topology-by-design in composite systems across wave physics.