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I am not sure whether these examples can be described as nonstandard, but they are among my favorites. Ex 5 may be too advanced for your course.

1. Proof of Poncelet's porism using correspondences and the group law on an elliptic curve.

2. There are 24 inflexion lines and 28 bitangent lines to a smooth plane quartic (Clemens, Scrapbook of complex curve theory).

3. Configuration of lines on a smooth del Pezzo surface of degree $d$ obtained by blowing up $n=9-d$ points in $\mathbb{P}^2$ is described via the root system of type $E_n$ (Manin, Cubic forms).

4. Configuration of exceptional curves on the resolution of a Klein-du Val singularity is given by a Dynkin graph (Shafarevich, Basic algebraic geometry, vol 1).

5. How many conics in $\mathbb{P}^2$ touch 5 conics in general position? (Fulton, Intersection theory)