Is any type of geometry $not$ "infinitesimally Euclidean"?

I know this is an old question, but I just discovered it, and I'm a little confused by the responses from others, because there is an obvious problem with the following claim you make:

Any geometry of (smooth) manifolds seems to be infinitesimally Euclidean, even for those without a Riemannian metric, since each neighborhood is (diffeomorphic) homeomorphic to Euclidean space.

You are confusing the category of smooth manifolds with the category of Riemannian manifolds, or the subject of differential topology with the subject of differential geometry.

More specifically, it makes no sense to talk about “geometry” in the category of mere smooth manifolds (even though we often learn about the apparatus of bare smooth manifolds in courses called “differential geometry”). The “geometry” is defined by the Riemannian metric and is something over and above the bare smooth structure. There is thus no sense at all to the statement that “the geometry of any smooth manifold is infinitesimally Euclidean”: there’s no geometry to speak of in the first place! Mere diffeomorphisms don’t capture, transport, or preserve geometry; isometries do.

So yes, every smooth manifold is locally diffeomorphic to Euclidean space, and yes this has nothing to do with whether or not the manifold is endowed with a metric structure. There are thus no local invariants in differential topology, for the reason you state: all objects in the category of smooth manifolds are locally diffeomorphic. But there are local invariants in differential geometry; not all objects in that category are locally isometric.

The subject of topology, and its sub-subject metric spaces, contains many examples of "absolute geometry" spaces that are not infinitesmally Euclidean, neither in the metric sense, nor the smooth sense, nor the topological sense. These subjects abound with many natural examples.

For example, infinite dimensional Hilbert spaces, which have many examples amongst function spaces, are not locally homeomorphic to Euclidean space. And yet their geometry is of intense interest. Take a look at any functional analysis book.

For another example, there is an entire theory devoted to metric spaces of nonpositive curvature. See the book of Bridson and Haefliger for a better feel.

Riemann in his famous essay of 1854 considered only metrics that are infinitesimal. For this reason Riemannian geometry properly speaking is only concerned with this type of manifold. A generalisation is known as Finsler spaces. Here infinitesimally the space looks like a Banach space, which is more general than Euclidean space.