From Neil Sloan's video Dungeon Numbers (extra) - Numberphile, the value for the infinite dungeon $D(x)$ is defined as the limit of the sequence
$$ f_{n+1} = (x)_ {f_n} $$
where $(\cdot)_b$ is the number interpreted in base $b$. As an example, he computes the sequence for $x = 1.1$ which remarkably equals $\phi$, the golden ratio. Indeed, this is easy to verify numerically:
$$ \begin{array}{} f_0 = 1.1 \\ f_1 = 1.90909090909217228205684753741421575\\ f_2 = 1.52380952380917721966240714083178116\\ f_3 = 1.65625000000014926379773289615036607\\ f_4 = 1.60377358490560596435426184696217646\\ f_5 = 1.62352941176472703752648837052858502\\ f_6 = 1.61594202898549922043011560202282156\\ f_7 = 1.61883408071749186281905685768538552\\ f_8 = 1.61772853185595450582692828879807295\\ f_9 = 1.61815068493150729747058805145095582\\ \cdots \end{array} $$
It's also worth noting that the $D(x)$ is only interesting in the range $x \in [1,10]$, small values vanish and large values do not converge. Out of curiosity, I plotted the values of $D(x)$ over the range $[1, 2]$ at the discrete interval of $0.001$:
It looks like there is a pattern of self-similarity here so I tried a finer resolution using the interval of $0.0001$
I'm not sure how to quite handle this function (it looks non-smooth, but I don't even know if it is continuous!). Is there a way to determine if the function is self-similar in some way?
For reference, each point on the curve was computed using 200 digits of precision and truncated at $f_{80}$. These numbers were empirically chosen as I didn't see a difference at lower values.