Connections: Exponential Map Given a smooth manifold.
Suppose it has an (affine) connection.
How is the exponential map constructed?

The usual construction of the exponential map in Riemannian geometry works also for a  general  affine connection, even if it does not come from any metric, as follows. 
Let $\nabla$ be an affine connection  on some manifold $M$, ie a connection on the tangent bundle of $M$. 
A parametrized curve in $M$ is called a geodesic if its tangent is parallel with respect to  $\nabla$. The equation for a geodesic, written in coordinates, is a 2nd order ODE, hence   given any initial point $p\in M$ and $v\in T_pM$, there exists a unique geodesic $\gamma (t),$ defined for some open interval around $t=0$, such that $\gamma(0)=p$, $\gamma'(0)=v$. Define  $exp(v)=\gamma(1)$ (if $\gamma(t)$ is defined for $t=1$). Then $exp$ maps some open neighborhood of the origin in $T_pM$ to $M$. 
For a connection on an arbitrary vector bundle on $M$ I do not know of a definition of an exponential map (and I doubt there is). 
