Definition ("prefix-complete"): A sequence of positive integers $(a_n)_{n=1,2,3,\dots}$ will be called prefix-complete in base $b$ iff, for any positive integer $p$, there is some $a_n$ whose base-$b$ digits begin with those of $p$. (Concatenating the digits of all the terms of a prefix-complete sequence would therefore produce a disjunctive sequence -- one in which all finite digit-strings occur as subwords.)
Question: For which hyperoperations $\circ\in({\tt Succ},+,\times,\uparrow,\uparrow\uparrow,\dots)$ is the sequence $(2\circ n)_{n=1,2,3,\dots}$ prefix-complete in base ten?
It's easy to see that prefix-completeness holds for the first three hyperoperations: all the required prefixes clearly occur for $(2\ {{\tt Succ}}\ n)_n = (1+n)_n$ and for $(2+n)_n$, while $(2\times n)_n$ is just the even positive integers (whose numerals include, e.g., every required prefix followed by a $0$).
In fact, Istrate & Paun proved that any strictly increasing sequence $(a_n)_n$ of positive integers is prefix-complete in every integer base $b\ge 2$ if $\lim_\limits{n\to\infty}(a_{n+1}/a_n)=1.$
Now for $a_n=2\uparrow n,$ however, $\lim_\limits{n\to\infty}{2\uparrow(n+1)\over 2\uparrow n}\ne 1;$ nevertheless, $(2\uparrow n)_n$ is indeed prefix-complete. This follows from proofs (e.g., here) appealing to the fact that, because $\log_{10}(2)$ is irrational, the fractional parts $\{n\log_{10}(2)\}$ are dense in the real interval $[0,1].$
Does anyone see an approach for the case of tetration, etc? Can it be shown that the fractional parts $\{(2\uparrow\uparrow n)\log_{10}(2)\}$ are dense in $[0,1]?$
I can't even see a way to prove that there exists some tower of $2$s that begins with, say, a $9$: $${{2^{2^{\cdot^{\cdot^{2}}}}}}=(9\dots)_{10}$$
NB: Letting $S_k:=(2\circ_k n)_{n=1,2,3,\dots}$ where $(\circ_0,\circ_1,\circ_2,\circ_3,\dots)=({\tt Succ},+,\times,\uparrow,\dots),$ we note that $S_k$ is a subsequence of every $S_j$ with $2\le j\le k;$ hence, if some $S_k$ is not prefix-complete in a given base, then neither is any $S_j$ with $j\gt k.$ Therefore, prefix-completeness will either hold for all the sequences $(S_0,S_1,S_2,\dots),$ or it will hold only for an initial finite segment $(S_0,S_1,\dots,S_m)$ where $m\ge 3.$
