First step to \(\omega^\omega\)[]
What one fool can do, another can.
—Ancient Simian Proverb
Growth rate: \(\omega^{\omega}\)
1 = A
2 = B
3 = AA
3+1 = AA.A
3+2 = AA.B
3+3 = (AA)(AA)
3+3+1 = (AA)(AA).A
3+3+2 = (AA)(AA).B
3+3+3 = (AA)(AA)(AA)
3^2 = BB
3^2+1 = BB.A
3^2+2 = BB.B
3^2+3 = BB.AA
3^2+3+1 = BB.AA.A
3^2+3+2 = BB.AA.B
3^2+3+3 = BB.(AA)(AA)
3^2+3^2 = (BB)(BB)
3^2+3^2+1 = (BB)(BB).A
3^2+3^2+2 = (BB)(BB).B
3^2+3^2+3 = (BB)(BB).AA
3^2+3^2+3+1 = (BB)(BB).AA.A
3^2+3^2+3+2 = (BB)(BB).AA.B
3^2+3^2+3+3 = (BB)(BB).(AA)(AA)
3^2+3^2+3+3+1 = (BB)(BB).(AA)(AA).A
3^2+3^2+3+3+2 = (BB)(BB).(AA)(AA).B
3^3 = AAA