Binary pulsar observations and gravitational wave detections seriously constrained scalar-tensor theories with massless scalar field allowing only small deviations from general relativity. If we consider a nonzero mass of the scalar field, though, significant deviations from general relativity are allowed for values of the parameters that are in agreement with the observations. In the present paper we extend this idea and we study scalar-tensor theory with massive field with self-interaction term in the potential. The additional term suppresses the scalar field in the neutron star models in addition to the effect of the mass of the scalar field but still, large deviations from pure GR can be observed for values of the parameters that are in agreement with the observations.