I guess I'm a little confused about one of your criteria ("without . . . acceleration"): in this example, the object should always experience a non-zero acceleration because the $\sim1/r^2$ relation is only 0 at $r\to\infty$.

HDE 226868 you are correct i will edit

And i have something else in my mind lets just say the initial acceleration is 0.

If the initial acceleration is zero, then the initial force is zero, and therefore the particle has infinite distance from the central mass (or in other words, there does not exist any place in space where the acceleration is zero — unless we take other gravitating bodies into account). More generally, for any time-independent field, if both the velocity and the acceleration at the beginning are zero, it follows that the particle stays there forever (unless disturbed by an external force).

It is an exercise the gravitational field exists only for some r below a limit and a test particle enters somehow(pops into existence) at the edge of the field.