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This version can go above R(10^30) since space consumption is irrelevant and it can simply be left to run some more, but above R(10^30) the fans go into jet engine mode and I do not want to stress my potato. Here is the data table showing the exponent k, the value of R(10^k), the bit count of the R value and the number of generators:
$ cat levels.txt
1 69 7 0
2 5764 13 1
3 526334 20 2
4 50874761 26 3
5 5028514748 33 3
6 500918449417 39 3
7 50029364025646 46 3
8 5000934571821489 53 4
9 500029664631299282 59 4
10 50000940106333921296 66 4
11 5000029765607020319048 73 4
12 500000941936492050650505 79 4
13 50000029798618763894670256 86 4
14 5000000942529842698007077786 93 4
15 500000029809255627825531266625 99 5
16 50000000942720152238624116795401 106 5
17 5000000029812655507343465281696595 112 5
18 500000000942780823112495107784753816 119 5
19 50000000029813737262126730811322149673 126 5
20 5000000000942800098290022420982686040347 132 5
21 500000000029814080548392288266955229183571 139 5
22 50000000000942806209878293665994446398371544 146 5
23 5000000000029814189323670710814676032031444555 152 5
24 500000000000942808145472258657037814775197247031 159 5
25 50000000000029814223760934912839828327249688721190 166 5
26 5000000000000942808758091081952084125868709915711089 172 5
27 500000000000029814234658067055252766458087914346630413 179 5
28 50000000000000942808951913512756782469229223199209677641 186 5
29 5000000000000029814238105320163714566005337846445873165675 192 5
30 500000000000000942809013222649956573426148951590151576716407 199 6It can be seen that the number of generators increases by 1 roughly when k doubles, which corresponds to squaring n. This backs up the space consumption as O(log(log(n))). For the next version I might incorporate >>33's complement idea to see what constant factor improvement it yields, or I might add third-order S-groups which will bring the runtime down to O(n^0.125).
Please verify the value of R(10^30) with other implementations.