Descending Dungeons and Iterated Base-Changing

For real numbers a, b > 1, let ab (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 nth terms 10 11 12 . . . (n− 1) n or n (n− 1) . . . 12 11 10 grow roughly like 1010n log log n , where the logarithms are to the base 10. We also investigate the behavior as n increases of the sequence a a a . . . a, with n a’s, parenthesized from the bottom upwards. This converges either to a single number (e.g. to the golden ratio if a = 1.1), to a two-term limit cycle (e.g. if a = 1.05) or else diverges (e.g. if a = 100 99 ).