I should say, you can absolutely approximate the orbits of n bodies under classical gravitation, and its generally pretty easy to do so, and we do it a lot for very large n. And we can write down iterative approximations that provably converge to the true motion (so its not like, yeah this looks legit, but it starts to deviate in the 150th digit in some way that becomes dominant after a billion years - though chaos is an issue, there's stuff like KAM theory that gives a semi-statistical description that works as long as the orbits are stable-ish).

Its just that, we can't write down some finite expression of known transcendental and algebraic functions that's just 'this is the orbits, for all time' for all possible initial values. It's always going to have a limit, or an infinite sum, or an integral.