Descending Dungeons and Iterated Base-Changing

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For real numbers a, b> 1, let as ab 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 abcd...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 10₁1₁2...(n-1)ₙ or n(n-1)...₁2₁1₁0 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 aₐₐ...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 = frac10099).