We compute the three-point correlation function of primordial scalar density perturbations in a general single-field inflationary scenario, where a scalar field φ has a direct coupling with the Ricci scalar R and the Gauss-Bonnet term G. Our analysis also covers the models in which the Lagrangian includes a function non-linear in the field kinetic energy X = −(φ)²/2, and a Galileon-type field self-interaction G(φ,X)□φ, where G is a function of φ and X. We provide a general analytic formula for the equilateral non-Gaussianity parameter f_NL^equil associated with the bispectrum of curvature perturbations. A quasi de Sitter approximation in terms of slow-variation parameters allows us to derive a simplified form of f_NL^equil convenient to constrain various inflation models observationally. If the propagation speed of the scalar perturbations is much smaller than the speed of light, the Gauss-Bonnet term as well as the Galileon-type field self-interaction can give rise to large non-Gaussianities testable in future observations. We also show that, in Brans-Dicke theory with a field potential (including f(R) gravity), f_NL^equil is of the order of slow-roll parameters as in standard inflation driven by a minimally coupled scalar field.