In his original paper Goodstein (1947) [6] introduces a hierarchy of functions. The third function of that hierarchy, usually referred to as the Goodstein function, is shown to have the same growth rate as , the th function of the Hardy hierarchy. More generally, the question is raised whether there is a clear connection between larger ordinals on the one hand and the so-called level- Goodstein sequences, for , and their termination on the other. In other words, we try to measure the growth rate of level- Goodstein functions by relating them to a standard ordinal-indexed hierarchy of functions, in this case the Hardy hierarchy. In this paper, we define appropriate Goodstein sequences for prominent ordinals up to the Bachmann–Howard ordinal.