A spin-1/2 triangular-lattice Heisenberg antiferromagnet (TLHAF) is a prototypical frustrated quantum magnet, which exhibits remarkable quantum many-body effects that arise from the synergy between spin frustration and quantum fluctuation. The ground-state properties of a spin-1/2 TLHAF are theoretically well understood. However, the theoretical consensus regarding the magnetic excitations is limited. The experimental study of the magnetic excitations in spin-1/2 TLHAFs has also been limited. Here we show the structure of magnetic excitations in the spin-1/2 TLHAF Ba3CoSb2O9 investigated by inelastic neutron scattering. Significantly different from theoretical expectations, the excitation spectrum has a three-stage energy structure. The lowest-energy first stage is composed of dispersion branches of single-magnon excitations. The second and third stages are dispersive continua accompanied by a columnar continuum extending above 10 meV, which is six times larger than the exchange interaction J = 1.67 meV. Our results indicate the shortcomings of the current theoretical framework.Two-dimensional frustrated magnets are heavily studied because theories predict that quantum effects may lead to the emergence of fractionalized excitations. Ito et al. use inelastic neutron scattering to show that the excitation spectrum of Ba3CoSb2O9 disagrees with current theoretical expectations.