  Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct
a group with uncountably many pairwise non-homeomorphic asymptotic cones. In
this paper we define a class of metric spaces which display a wide range of
behaviors with respect to iterated asymptotic cones, and we use those to
construct examples within the class of groups. Namely, we will show that there
exists a group whose iterated cones are pairwise non-homeomorphic, or
periodically homeomorphic.
