The term degenerate is used to describe abelian varieties whose Hodge rings contain exceptional cycles -- Hodge cycles that are not generated by divisor classes. We can see the effect of the exceptional cycles on the structure of an abelian variety through its Mumford-Tate group, Hodge group, and Sato-Tate group. In this talk I will discuss degeneracy through these different but related lenses, specializing to Jacobians of hyperelliptic curves of the form $y^2=x^m−1$. Together, we will explore the various forms of degeneracy for several examples, each illustrating different phenomena that can occur.