A power tower of a number $x$ is typified by

$$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$

Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers

$$x_1^{x_2^{x_3^{ \cdots\cdots^{x_k}}}},$$

where each $x_h$ is $2$ or $3,$ and $k \geq 2.$ Let $T_2$ be the subset of $T$ consisting of towers rising from $x_1=2.$ Let $R$ be the sequence of ranks of towers in $T_2$ when all the towers in $T$ are jointly ranked.

The first $15$ terms of $R$ are $$1, 2, 4, 7, 8, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29.$$ What are the next terms?

Note that $T$ can be obtained recursively from $t_2 = \{2^2,2^3,3^2,3^3\}$ by defining

$$t_n =2^{t_{n-1}} \cup 3^{t_{n-1}}$$

for $n \geq 3;$ then $T$ is the union of the sets $t_n$ for $n \geq 2.$

For a top-first version of the problem, change $x_1=2$ to $x_k=2,$ where $k$ is the height of the tower. Then the first $17$ terms are $$1,3,4,6,10,11,12,15,16,19,20,23,24,25,26,27,28,\ldots.$$ Here, too, the question is: what are the next terms?