Dison and Riley's hydra groups $G_k$ contain certain extravagantly distorted subgroups $H_k$; the function measuring the
distortion grows like the k-th Ackermann function. One wants to know if finite quotients can distinguish elements that are not in $H_k$, as a positive answer would allow you to construct an elementary family of finitely presented, residually finite groups with fast-growing Dehn functions. (Kharlampovich, Mysanikov, and Sapir constructed a family of groups with these properties in 2012, but their examples are difficult.) I'll explain why we get a negative answer and the types of weaker solutions we might hope for.