Optical precision spectroscopy of isotope shifts can be used to test for new forces beyond the standard model, and to determine basic properties of atomic nuclei. We measure isotope shifts on the highly forbidden 1/2S2 → 7/2F2 octupole transition of trapped 168 ,170 ,172 ,174 ,176Yb ions. When combined with previous measurements in Yb+ and very recent measurements in Yb, the data reveal a King plot nonlinearity of up to 240 σ . The trends exhibited by experimental data are explained by nuclear density functional theory calculations with the Fayans functional. We also find, with 4.3 σ confidence, that there is a second distinct source of nonlinearity, and discuss its possible origin.