It is a cute thoerem of Franz-Erick Wolter that a complete $n$-dimensional Riemannian manifold $M$ is necessarily diffeomorphic to $\mathbb R^n$ if there is a point $p\in M$ such that the function $d(p,\mathord-):M\to\mathbb R$ has directional derivatives at all points and in all directions. This provides examples.

See [Wolter, Franz-Erich. Distance function and cut loci on a complete Riemannian manifold. Arch. Math. (Basel) 32 (1979), no. 1, 92--96. MR0532854]

It is a cute thoerem of Franz-Erick Wolter that a complete $n$-dimensional Riemannian manifold $M$ is necessarily diffeomorphic to $\mathbb R^n$ if there is a point $p\in M$ such that the squared-distance function $d(p,\mathord-)^2:M\to\mathbb R$ has directional derivatives at all points and in all directions. This provides examples.

See [Wolter, Franz-Erich. Distance function and cut loci on a complete Riemannian manifold. Arch. Math. (Basel) 32 (1979), no. 1, 92--96. MR0532854]