There are two badges, Tenacious and Unsung Hero that never have been awarded. And other than the Illuminator badge that with enough diligence eventually can be awarded to someone willing to put in the hours, I'm not even sure it is possible to honestly achieve the easier one of the two, Tenacious, which says:
Zero score accepted answers: more than 5 and 20% of total.
The entire site has just 136 accepted answers with 0 score out of 104K as of this writing, a bit more than 0.1%. It seems extremely unlikely anyone would get 20%+ such answers out of theirs unless they collaborated with someone to engineer just such an outcome.
Why do we still have these badges? Is this because we cannot retire badges that have no useful application?