[math-fun] continued subscripts

[Marc LeBrun]

Recent correspondence has inspired me that, analogous to stacking 
superscripts for iterated exponentiation, we may stack the subscripts 
that are vulgarly used as "base" notation.

For example, we may apprehend, among 239's many sterling properties, 
the following gratuity:

10
   11
     12
       13
         14
           15
             16                 = 239.
               17
                 18
                   19
                     20
                       21
                         22
                           23

That is, working up from the bottom:

...          ...        ...
    21     =     21    =    97  and so on  = 239.
      22           48
        23

Question: If stacking superscripts produces a "tower", what are 
stacked subscripts?  A "dungeon"?

The sequence following this incrementing pattern of subscripts, namely,

      10,   10,    10,    10,     ...
              11     11     11
                       12     12
                                13

has (if I've computed them correctly) the values

10 11 13 16 20 25 31 38 46 55 65 87 135 239 463 943 1967 4143 8751 18479 38959
103471 306223 942127 2932783 9153583 28562479 89028655 277145647 861652015
2675637295 10173443119 41132125231 168836688943 695134284847 2862242032687
11778342480943 48433422101551 199016451894319 817199416306735 3353334654921775
16351027752823855 82924577766468655 423717750455364655 2167310727003204655
11083411175259204655 56654591244123204655 289463880726363204655
1478277273827163204655 7546178967779163204655 38504861079779163204655
227971995605219163204655 1387065053878499163204655 8475364910242019163204655
51807613088766179163204655 ...

(which I will submit to the OEIS as soon as I can craft a concise 
explanation!).

Puzzle: the least-significant digits seem to be stabilizing ... why or why not?

Of course we can write the numerals in the stack in other bases, such 
as binary:

      10,   10,    10,     10,       ...
              11     11      11
                       100     100
                                  101

and these have as their first few values

2 3 5 26 1370 9840770 851566070376026 ...

but it grows rapidly--the next two terms have ~144 and ~450 digits 
respectively (and seem beginning to show a distinct tendency to end 
with the digits ...26 ?!)

Question: What is a good approximate formula for a(n)?

And of course other continued subscript patterns besides simply 
incrementing the terms are possible.