Coulomb interactions between the carriers may provide the mechanism for enhanced unconventional superconductivity in the copper oxides. However, they simultaneously cause inelastic quasiparticle scattering that can destroy it. Understanding the evolution of this balance with doping is crucial because it is responsible for the rapidly diminishing critical temperature as the hole density p is reduced towards the Mott insulating state. Here, we use tunnelling spectroscopy to measure the T→0 spectrum of electronic excitations N(E) over a wide range of hole density p in superconducting Bi"2Sr" 2CaCu"2O(8+δ). We introduce a parameterization for N(E) based on a particle-hole symmetric anisotropic energy gap Δ(k)≤Δ1(cos(kx) cos(ky))/2 plus an inelastic scattering rate that varies linearly with energy 2(E)≤αE. We demonstrate that this form of N(E) enables successful fitting of differential tunnelling conductance spectra throughout much of the Bi"2Sr"2CaCu"2O(8+δ) phase diagram. We find that Δ1 values rise with falling p along the familiar trajectory of excitations to the pseudogap energy, whereas the energy-dependent inelastic scattering rate 2(E)≤αE seems to be an intrinsic property of the electronic structure and rises steeply for p<16. Such diverging inelastic scattering may play a key role in suppression of superconductivity in the copper oxides as the Mott insulating state is approached. © 2008 Nature Publishing Group.