I have examples for $\ell_2$ distance in $\mathbb{R}^2$.

In particular, take $2n+2$ points at the vertices of a regular polygon of unit radius, and set the distance parameter to be slightly less than $2$ (so that two points are joined by an edge if and only if they are not antipodal).

The corresponding simplicial complex is the boundary of the $(n+1)$-dimensional orthoplex (hyper-octahedron), which is homeomorphic to $S^n$ and therefore has a non-trivial $H_n$.

Fun fact: you can take connected sums of these regular polygon examples to realise finite Vietoris-Rips complexes in $\mathbb{R}^2$ with any finitely-supported finite sequence of Betti numbers.

I have examples for $\ell_2$ distance in $\mathbb{R}^2$.

In particular, take $2n+2$ points at the vertices of a regular polygon of unit radius, and set the distance parameter to be slightly less than $2$ (so that two points are joined by an edge if and only if they are not antipodal).

Here's an image for $n=2$, from this page:

The corresponding simplicial complex is the boundary of the $(n+1)$-dimensional orthoplex (hyper-octahedron), which is homeomorphic to $S^n$ and therefore has a non-trivial $H_n$.

Fun fact: you can take connected sums of these regular polygon examples to realise finite Vietoris-Rips complexes in $\mathbb{R}^2$ with any finitely-supported finite sequence of Betti numbers.