What are some examples of incomplete nonextendible manifolds?

For a simple example, take the surface $C=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=z^2,z>0\}$ with the metric inherited from the Euclidean metric on $\mathbb{R}^3$. This is a cone without its vertex. To complete it as a metric space, you would just need to add in the vertex, but this completion is not a Riemannian manifold since it is not smooth at the vertex.

Although this has been answered a while ago, I cannot resist describing one of my favorite examples, which is related to the comment of @TedShifrin.

Start with the ordinary Euclidean metric on $\mathbb R^2 - \{(0,0)\}$ and then lift it to the universal cover. The resulting space is isometric to $(0,\infty) \times (-\infty,+\infty)$ with coordinates $r \in (0,\infty)$, $\theta \in (-\infty,+\infty)$, equipped with the metric $dr^2 + r^2 d\theta^2$. A nonconvergent sequence $(r_n,\theta_n)$ in this space is Cauchy if and only if $r_n \to 0$: indeed the infimal path length (aka distance) from $(r_1,\theta_1)$ to $(r_2,\theta_2)$ is $\le r_1 + r_2$. For similar reasons, all nonconvergent Cauchy sequences are equivalent. Thus the completion is obtained by adding one point.