Main Question or Discussion Point
I'm looking for a simple example of a differentiable manifold that doesn't have an associated riemann metric.
We have to be careful with definitions here. In particular, it seems to me that a non-second-countable smooth manifold may be equipped with a local Riemann metric but not be a metric space.The proof that every manifold has a metric (as well as the proof of Whitney's embedding theorem) relies on paracompactness. If you drop this requirement, you can have all sorts of aberrations.
In fact, if your space has a metric, it has to be second countable (delta-balls type argument).