We consider a version of the nonlinear Schroedinger equation (NLS) with point nonlinearity, which can formulated as the linear Schroedinger equation away from the spatial origin together with a nonlinear jump condition in the derivative across the origin. This model can be viewed as a limiting form of a concentrated nonlinearity and exhibits many of the same properties as the standard nonlinear Schroedinger equation. In fact, in most cases, the analysis is simpler than for standard NLS and thus simpler proofs and/or stronger results are possible. As an illustration, we construct self-similar finite energy blow-up profiles for the entire mass supercritical range. For the standard NLS, this has only been achieved in the 'slightly supercritical' region.