The Culler-Vogtmann Outer space $\text{CV}_n$ is an analogue of Teichmuller space for the group $\text{Out}(F_n)$.
Is there any notion of a geodesic path in $\text{CV}_n$? Are there different competing definitions of geodesic?
If so, what would be a simple example of a geodesic path vs. a non-geodesic one, say on $\text{CV}_2$?
Besides the geodesic paths of the asymmetric metric $d(\cdot,\cdot)$ that are mentioned in other answers (namely paths such that $d(\gamma(s),\gamma(t)) = t-s$ if $s \le t$), there is another class of paths with many uses known as Stallings fold paths. You can see some discusions of them in the outer space context, with applications, in these lecture notes of Bestvina, these notes of Kapovich and Myasnikov, and this issue of the AMS Memoirs by Handel and myself.
I would recommend looking at Karen Vogtmann's survey article On the geometry of Outer space, in the Bulletin of the AMS (and available online). The Lipschitz metric is defined and discussed in Section 3, and Section 5 discusses geodesics in this metric.
In particular, the following excerpt taken from pages 37-38 contains an example of two directed geodesics in $\text{CV}_2$ between the same two endpoints in adjacent cells. (As mentioned in @andyputman's answer, the Lipschitz metric is not symmetric so geodesics generally depend on the direction of travel.) In this example, any path between the two endpoints which crosses the boundary line between the cells at a different point would not be geodesic.
To talk about geodesics, you need a notion of distance. For outer space, something strange happens: there is a natural notion of distance (called the "Lipshitz metric"), but it is not symmetric. In other words, there exist points $x$ and $y$ in Outer space such that $d(x,y)$ and $d(y,x)$ are different! Nonetheless, one can still talk about geodesics.
For an introduction to this circle of ideas, I recommend Bestvina's Park City notes:
Bestvina, Mladen, Geometry of outer space. Geometric group theory, 173–206, IAS/Park City Math. Ser., 21, Amer. Math. Soc., Providence, RI, 2014.
The whole set of notes is useful, but Lecture 3 is where the distance function is defined.
You should consult the oeuvre of Yael Algom-Kfir, in particular "Strongly Contracting Geodesics in Outer Space", whereupon enlightenment will ensue.
