Let $M$ be a smooth manifold, and let $\nabla$ denote a connection on $M$.
Question: If $M$ is compact, is every maximal geodesic of $\nabla$ defined for all $t\in \mathbb{R}$?
I know that, if $M$ is a Riemannian manifold and $\nabla$ its Levi-Civita connection, then the answer is yes. Maybe the proof sketched in this answer could work in this case too?
Consider $M$ the quotient of $\mathbb{R}^n-\{0\}$ by $h(x)=2x$ it is a compact manifold. Since thw Eucledean connection of $\mathbb{R}^n$ is preserved by $h$, it induces a connection on $M$ which is not complete. To see this consider a geodesic the image of the geodesic $(t,0...0), t\in (0,1]$.