The first full amplitude analysis of B+ -> J/psi phi K+ with J/psi -> mu(+)mu(-), phi -> K+K- decays is performed with a data sample of 3 fb(-1) of pp collision data collected at root s = 7 and 8 TeV with the LHCb detector. The data cannot be described by a model that contains only excited kaon states decaying into phi K+, and four J/psi phi structures are observed, each with significance over 5 standard deviations. The quantum numbers of these structures are determined with significance of at least 4 standard deviations. The lightest has mass consistent with, but width much larger than, previous measurements of the claimed X(4140) state. The model includes significant contributions from a number of expected kaon excitations, including the first observation of the K*(1680)+ -> phi K+ transition.