Each site has a list of badges with the number of times they've been awarded.
Not trying to brag, but on this site, i.e. Meta Stack Exchange, the only badge which is awarded only 1 timethe only badge which is awarded only 1 time is Illuminator. Tenacious and Unsung Hero could theoretically be awarded here but there's too much voting for the amount of questions here for those badges to be practically feasible.
Illuminator is also the rarest badge in the entire network, at least among the ones which can be obtained on every site.