Dear Qiaochu, if $A$ is a discrete valuation ring and if $B$ is an étale algebra over $A$, then $B$ is a discrete valuation ring. In a related vein, the henselization of a discrete valuation ring $A$ is a discrete valuation ring $A^h$ (however it is not étale over $A$, for example because it is not finitely generated ).If $A$ is the local ring of a point on a curve in the Zariski topology, then $A^h$ is the local ring of that point in the étale topology.
A very concrete example: the henselization of the local ring $A=\mathcal O_{\mathbb A^1,0}$ of the complex affine line at the origin is the subring of the ring of formal series $\mathbb C [[T]]$ consisting of those series that are algebraic over $A$.
These seem to be examples not on your list, but I'll let you be the judge of their exotism....