Let $\nabla$ be a torsionfree affine connection on $\mathbb{R}^n$ which is invariant under the group of translations in $\mathbb{R}^n$.
What are necessary and sufficient conditions on $\nabla$ for it to be geodesically complete?
If the previous question is too difficult to settle, perhaps the following one is easier. What is a class (or maybe classes) of geodesically complete such $\nabla$s?
One class of translation invariant complete affine connections, in the plane, was discovered by Yeaton Clifton. These are the translation invariant affine connections in the plane whose geodesic flow preserves a translation invariant positive definite quadratic form. One can show (see the upcoming draft of my Introduction to Cartan geometries, which should be on the arxiv tomorrow), that there is a unique one which is not the standard Euclidean connection: the one whose geodesic equations are $$ 0=\ddot{x}+\dot{y}^2, $$ $$ 0=\ddot{y}-\dot{x}\dot{y}. $$ More generally, to get a translation invariant Riemannian metric to be preserved along the geodesic flow of $$ 0=\ddot{x}^i+\Gamma^i_{jk}\dot{x}^j\dot{x}^k $$ we can arrange, by linear transformation, that the Riemannian metric is the Euclidean metric, and differentiate $ds^2=(\dot{x}^i)^2$ to see that the connection has precisely to satisfy $$ 0=\Gamma^i_{jk}\dot{x}^i\dot{x}^j\dot{x}^k $$ i.e. the symmetrization of $\Gamma^i_{jk}$ in $ijk$ has to vanish. This ensures completeness clearly. It does not require that the connection be Levi-Civita, or even torsion free.