We investigate natural systems of fundamental sequences for ordinals below the Howard–Bachmann ordinal and study growth rates of the resulting slow growing hierarchies. We consider a specific assignment of fundamental sequences that depends on a non-negative real number $\varepsilon$. We show that the resulting slow growing hierarchy is eventually dominated by a fixed elementary recursive function if $\varepsilon$ is equal to zero. We show further that the resulting slow growing hierarchy exhausts the provably recursive functions of $\sfb{ID}_1$ if $\varepsilon$ is strictly greater than zero. Finally, we show that the resulting fast growing hierarchies exhaust the provably recursive functions of $\sfb{ID}_1$ for all non-negative values of $\varepsilon$. Our result is somewhat surprising since usually the slow growing hierarchy along the Howard–Bachmann ordinal exhausts precisely the provably recursive functions of $\sfb{PA}$. Note that the elementary functions are a very small subclass of the provably recursive functions of $\sfb{PA}$, and the provably recursive functions of $\sfb{PA}$ are a very small subclass of the provably recursive functions of $\sfb{ID}_1$. Thus the jump from $\varepsilon$ equal to zero to $\varepsilon$ greater than zero is one of the biggest jumps in growth rates for subrecursive hierarchies one might think of.