The shot noise power of a diffusive conductor in the presence of both strong inelastic electron-electron and electron-phonon scattering is calculated. It is shown that electron-electron inelastic scattering does not suppress shot noise strongly and the noise power is (√3/2)e|I¯|. In contrast to this, inelastic coupling with phonons affects the nonequilibrium current noise substantially leading to a non-linear deviation from standard shot noise formulae.