Chaos requires nonlinearity. A gravitational body's motion affects every other body, which in turn affects the original body, etc. So you can't analyze the parts of the system independently and then just add up the results. A body's motion indirectly affects its own future motion. Spacetime tells matter how to move and matter tells spacetime how to bend, as Wheeler said.

In two-body problems, this infinite recursion converges to a closed-form solution, which means we can just write down the solution to the differential equations that describe the system's state and predict it at any time in the future.

Unfortunately there are no true two-body orbits in the universe. And with 3 and greater numbers of bodies that pleasant closed-form convergence doesn't happen. You get sensitive dependence to initial conditions (SDIC), with faster-than-linear divergence, and you can't solve the differential equations: You have to integrate them numerically.

The time horizon over which this divergence happens for planets is long relative to the time horizons we care about, but it's still SDIC and it's faster than linear. The same thing happens with the weather (the "butterfly effect") but there the time horizon for divergence (the Lyapunov time) is short enough that we humans can easily notice it.

https://en.m.wikipedia.org/wiki/Lyapunov_time