 Eventual dominance in a fast-growing hierarchy

For the standard fundamental sequences of limit ordinals below epsilon zero, the index order corresponds to eventual, rather than necessarily pointwise, growth. For example, under $\omega [n]=n+1$ and successor iteration $n+1$ times, $F_{\omega }(1)=F_2(1)=7$ whereas $F_3(1)=2047$. Limit stages are diagonal choices, so a larger index can give a smaller value at small arguments.