As far as I’m aware, in Newton’s gravity only gravitational potential energy can add up. Forces can add up, but they can also cancel out each other.
In the framework of GR, do curvatures of spacetime of two (or more) bodies add up to more curvature? Would light curve around the center of two bodies orbiting around each other, producing Einstein ring? Or around two body system as a whole?
In general relativity, gravity is not a force but a result of the worldlines of objects being curved towards one another due to the mass-energy of the objects curving spacetime itself. This is described by Einstein's equation, which is given by (in natural units where $G=c=1$) $$ R_{\mu\nu}-\frac{R}{2}g_{\mu\nu} +\Lambda g_{\mu\nu} = 8\pi T_{\mu\nu} $$ Where on the left hand side, $R_{\mu\nu}$ is the Ricci tensor obtained from contracting the first and third indices of the full Riemann curvature tensor, which is itself given by a complicated nonlinear expression involving second derivatives of the metric $g_{\mu\nu}$. $R$ is the trace of the Ricci tensor, and $\Lambda$ is a cosmological constant that is only included for completeness. This side describes the curvature of spacetime.
On the right hand side, we have the energy-momentum tensor which describes distribution of mass-energy in the spacetime. Since the expressions on the left are nonlinear, one cannot just add two energy-momentum tensors to get the metric for the combined system (Indeed, the full two-body problem in general relativity has not been solved analytically!). But one can write the energy-momentum tensor for a system of $N$ bodies, and at least solve numerically for the metric. In this sense, the fields do "add up".
However, they can never add up in a way that creates a stronger curvature away from the sources, i.e. concentrations of mass-energy.
