Descending Dungeons and Iterated Base-Changing

For real numbers a, b > 1, let a b (also written as a b) denote the result of interpreting a in base b instead of base 10. We define "dungeons" (as opposed to "towers") to be numbers of the form a b c d ... e, parenthesized either from the bottom upwards (preferred) or from the top downwards. Among other things, we show that the sequences of dungeons with n th terms 10 11 12 ... (n − 1) n or n (n − 1) ... 12 11 10 grow roughly like 10 10 n loglog n