Cyanobacterial phycobilisome complexes absorb visible sunlight and funnel photogenerated excitons to the photosystems where charge separation occurs. In the phycobilisome complex of Synechococcus elongatus PCC 7942, phycocyanin protein rods that absorb bluer wavelengths are assembled on allophycocyanin cores that absorb redder wavelengths. This arrangement creates a natural energy gradient toward the reaction centers of the photosystems. Here, we employ broadband pump-probe spectroscopy to observe the fate of excess excitations in the phycobilisome complex of this organism. We show that excess excitons are quenched through exciton-exciton annihilation along the phycocyanin rods prior to transfer to the allophycocyanin cores. Our observations are especially relevant in comparison to other antenna proteins, where exciton annihilation primarily occurs in the lowest-energy chlorophylls. The observed effect could play a limited photoprotective role in physiological light fluences. The exciton decay dynamics is faster in the intact phycobilisome than in isolated C-phycocyanin trimers studied in earlier work, confirming that this effect is an emergent property of the complex assembly. Using the obtained annihilation data, we calculate exciton hopping times of 2.2-6.4 ps in the phycocyanin rods. This value agrees with earlier FRET calculations of exciton hopping times along phycocyanin hexamers by Sauer and Scheer.