Recall two connections on a manifold are said to be *projectively equivalent*, if they have the same geodesics.  You want to know what local diffeos $\mathbb{R}^n\mapsto\mathbb{R}^n$ preserve geodesics; that is, what flat metrics on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one.  I think your conjecture, that the only such metrics are those obtained by affine-linear transformations of the Euclidean metric, is correct.

Let's discuss this first locally, then globally.

**Local question:** *What flat connections on a neighbourhood of $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?*

**Answer:**  If $n=1$, all.  If $n\geq 2$, exactly those obtained by what depending on your terminology you might call a a perspective transformation or a projective linear transformation or something else.

**Comment:**  Sketch proof below.  The case distinction (which explains your colleague's observation) comes from a factor of $1/(n-1)$ in the formula for the appropriate Schouten curvature tensor.  This is analogous to the case distinction $n\leq 2$ vs $n\geq 3$ in conformal geometry, see
http://en.wikipedia.org/wiki/Liouville's_theorem_(conformal_mappings)

Now we can deal with

**Global question:**  *What flat connections on $\mathbb{R}^n$ are projectively equivalent to the Euclidean one?*

**Answer:**  Just the standard one.  The others "blow up"/"go off to infinity"/involve-division-somewhere-by-zero if you try to extend them to all $\mathbb{R}^n$.



**Sketch proof of local version:**  Use the notation and some formulae from, eg.,
http://www.maths.adelaide.edu.au/michael.eastwood/projective.pdf

Let $\nabla$ be the standard connection on $\mathbb{R}^n$.  Consider a projectively equivalent connection defined by a 1-form $\Upsilon_i$ (so the new connection acts on a 1-form $\omega_i$ by,
$\nabla_i\omega_j - \Upsilon_i\omega_j -\Upsilon_j\omega_i$.)
The projective Schouten curvature tensor of $\nabla$ is zero (since it's flat).  If the new connection's also flat, then, applying appropriate transformation laws, we have that
$\nabla_i\Upsilon_j = \Upsilon_i\Upsilon_j$,
and that $\Upsilon_i$ is closed, so locally exact.

Write $\Upsilon = df$.  Then in standard Euclidean co-ordinates we have the system of PDE
$\partial_i\partial_j f = \partial_i f \ \partial_j f$
for the function $f$, which we can solve to get
$f(x^1, ... x^n) =-\log (a_1 x^1+ \cdots + a_nx^n + c)$
for some fixed constants $a_i$ and $c$.  Hence $\Upsilon_i = \frac{-a_i}{a_1 x^1+ \cdots + a_nx^n + c}$.  It should probably turn out that the family of connections this gives are all indeed flat, and correspond to the projective-linear-transformations of the Euclidean metric on $\mathbb{R}^n$.