The propagation of strongly nonlinear TM surface-guided waves in media controlled by a self-focusing third-order optical, unsaturated nonlinearity is analysed theoretically. In this theory both the amplitude and the wave number of the guided wave are power dependent. The reasons for studying these waves are explained and it is emphasised that strong nonlinearity can lead to a number of device applications. Although such devices are not simulated here, a number of planar structures that are of potential device interest are discussed. A mathematical theory is presented that is both free of previous approximations and capable of dealing with many types of nonlinear mechanisms. The waveguide structures that are considered are a single interface between semi-infinite linear and nonlinear media, a linear guide symmetrically bounded by self-focusing nonlinear media and two linear waveguides coupled through a nonlinear medium. The exact theory quickly leads to equations that can only be solved numerically but some degree of analytical progress is reported here for the single-interface case. Two methods of solution are presented, one based on the first integral of the nonlinear wave equation and the other based on a spline collocation numerical method. Full numerical results are given in the form of power variation with effective wave index and a number of sample field distributions. The results are especially interesting in connection with the structures that possess negative dielectric components and the superwaves of the nonlinearly coupled guides. In the former, maxima, minima and negative power regions arise. In the latter, for a critical coupling distance, there exists a region of guide index for which the power flow down the guide is approximately constant