  We study single, double and higher-order nonlinear Compton scattering where
an electron interacts nonlinearly with a high-intensity laser and emits one,
two or more photons. We study, in particular, how double Compton scattering is
separated into one-step and two-step parts, where the latter is obtained from
an incoherent product of two single-photon emissions. We include all
contributions to double Compton scattering and show that the exchange term,
which was not calculated in previous constant-crossed field studies, is in
general on the same order of magnitude as the other one-step terms. Our
approach reveals practically useful similarities between double Compton
scattering and the trident process, which allows us to transfer some of our
previous results for trident to double Compton scattering. We provide a new
gluing approach for obtaining the dominant contribution to higher-order Compton
scattering for long laser pulses. Unlike the standard gluing approach, our new
approach does not require the intensity parameter $a_0$ to be much larger than
one. For `hard' photons we obtain several saddle-point approximations for
various field shapes.
