Terence Tao's <a href="http://terrytao.wordpress.com/2012/04/02/a-cheap-version-of-nonstandard-analysis/">cheap nonstandard analysis</a> can be interpreted as taking place in a topos related to the topos $\text{Set}^{\mathbb{N}}$ of sets indexed by the natural numbers; see <a href="https://math.stackexchange.com/questions/454393/does-cheap-nonstandard-analysis-take-place-in-a-topos">this math.SE question</a> for details.