TITLE: Intrinsic metric with no geodesics
QUESTION [23 upvotes]: It seems that I have the needed example, but I want it to be simple and self-explaining...

Construct a nontrivial complete metric space $X$ with intrinsic metric which has no nontrivial minimizing geodesics.

Definitions:

A metric $d$ is called intrinsic if for any two points $x$, $y$ and any $\epsilon>0$ there is an $\epsilon$-midpoint $z$; i.e. $d(x,z),d(z,y)<\tfrac12 d(x,y)+\epsilon$. 
A minimizing geodesic is nontrivial if it connects two distinct points.
A meric space is nontrivial if it contains two distinct points.

Comments:

Clearly, $X$ can not be locally compact.

REPLY [3 votes]: This does not answer the question. The following paper contains a very natural example, namely the regular Frechet Lie group $Diff_{\mathcal S}(\mathbb R)$ of all diffeomorphisms of the real line which fall rapidly towards the identity, with a weak right invariant Riemannian metric induced by inner product
$$
G_{Id}(X,Y) = \int_{\mathbb R} X'Y'\,dx
$$
on the Lie algebra, which has the following property: 

Geodesic distance is an intrinsic metric, but there does not exist a single nontrivial geodesic. But it is not complete as a metric space.

This group is a normal subgroup (isometrically contained for the geodesic distance, by thm 4.5) in the slightly larger regular Lie group 
$Diff_{\mathcal S_1}(\mathbb R)$ where one allows shifts at $+\infty$. 
The inner product extends to the larger space of vector fields in the Lie algebra. The resulting weak right invariant Riemannian metric  (see 4.3) is flat,
has minimal geodesics between any two points (it is geodesically convex), it allows for a formula for geodesic distance, and it has a nice geodesic completion which is a monoid.
The geodesic equation is the Hunter-Saxton PDE in the real line. 
Any geodesic in  $Diff_{\mathcal S_1}(\mathbb R)$ hits the subgroup $Diff_{\mathcal S}(\mathbb R)$ at most twice.
Geodesic distance on $Diff_{\mathcal S}(\mathbb R)$ is an intrinsic metric; this follows from the proof of theorem 4.5.

Martin Bauer, Martins Bruveris, Peter W. Michor: Homogeneous Sobolev metric of order one on diffeomorphism groups on the real line. Journal of Nonlinear Science 24, 5 (2014), 769-808 (pdf)