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A fast-growing hierarchy indexes number-theoretic functions by an effective system of ordinal notations. Successor stages iterate the preceding function, while limit stages diagonalize along a chosen fundamental sequence of a limit ordinal. These choices are part of the definition. With the standard Cantor normal form sequences below epsilon zero, each fixed $F_{\alpha }$ for $\alpha <{\epsilon }_0$ is provably total in Peano arithmetic, but the uniform endpoint $F_{{\epsilon }_0}$ is not. Every function provably total in Peano arithmetic is eventually bounded by some lower level. These classification results are proved in epub.ub.uni-muenchen.de/3843/1/3843.pdf .