This is inspired by a math.SE question, where an infinite sequence of **pairwise distinct** natural numbers $a_1=1, a_2, a_3, ...$ has been defined as follows:

*$a_n$ is the smallest number such that $s_n:=\sqrt{a_n+\sqrt{a_{n-1}+\sqrt{\cdots+\sqrt{a_1}}}}$ is an integer.*

It turns out that this sequence, which BTW is still not in the OEIS, is in fact a permutation of $\mathbb N$. Moreover, the images show that both $(a_n)$and $(s_n)$ exhibit an interesting self-similarity, with two alternating structures and the ratio converging rapidly towards, as it seems, $3+2\sqrt2=(\sqrt2+1)^2\approx 5.828427$ (see this other answer). Below I have displayed the first $632$ entries in a way that makes it easy to see which numbers generate what is perceived on the pictures as lines.

```
1,
3, 2,
7, 6,
13, 5,
22, 4,
33, 10, 12, 21, 11,
32, 19, 20,
31, 30,
43, 9, 45, 18, 44, 29,
58, 8, 60, 17, 59, 28,
75, 16, 76, 27,
94, 15, 95, 26,
115, 14, 116, 25,
138, 24,
163, 23,
190, 35, 42, 57, 41, 74, 40, 93, 39, 114, 38, 137, 37, 162, 36,
189, 50, 56, 73, 55, 92, 54, 113, 53, 136, 52, 161, 51,
188, 67, 72, 91, 71, 112, 70, 135, 69, 160, 68,
187, 86, 90, 111, 89, 134, 88, 159, 87,
186, 107, 110, 133, 109, 158, 108,
185, 130, 132, 157, 131,
184, 155, 156,
183, 182,
211, 34, 218, 49, 217, 66, 216, 85, 215, 106, 214, 129, 213, 154, 212, 181,
242, 48, 248, 65, 247, 84, 246, 105, 245, 128, 244, 153, 243, 180,
275, 47, 281, 64, 280, 83, 279, 104, 278, 127, 277, 152, 276, 179,
310, 46, 316, 63, 315, 82, 314, 103, 313, 126, 312, 151, 311, 178,
347, 62, 352, 81, 351, 102, 350, 125, 349, 150, 348, 177,
386, 61, 391, 80, 390, 101, 389, 124, 388, 149, 387, 176,
427, 79, 431, 100, 430, 123, 429, 148, 428, 175,
470, 78, 474, 99, 473, 122, 472, 147, 471, 174,
515, 77, 519, 98, 518, 121, 517, 146, 516, 173,
562, 97, 565, 120, 564, 145, 563, 172,
611, 96, 614, 119, 613, 144, 612, 171,
662, 118, 664, 143, 663, 170,
715, 117, 717, 142, 716, 169,
770, 141, 771, 168,
827, 140, 828, 167,
886, 139, 887, 166,
947, 165,
1010, 164,
1075, 192, 210, 241, 209, 274, 208, 309, 207, 346, 206, 385, 205, 426, 204, 469, 203, 514, 202, 561, 201, 610, 200, 661, 199, 714, 198, 769, 197, 826, 196, 885, 195, 946, 194, 1009, 193,
1074, 223, 240, 273, 239, 308, 238, 345, 237, 384, 236, 425, 235, 468, 234, 513, 233, 560, 232, 609, 231, 660, 230, 713, 229, 768, 228, 825, 227, 884, 226, 945, 225, 1008, 224,
1073, 256, 272, 307, 271, 344, 270, 383, 269, 424, 268, 467, 267, 512, 266, 559, 265, 608, 264, 659, 263, 712, 262, 767, 261, 824, 260, 883, 259, 944, 258, 1007, 257,
1072, 291, 306, 343, 305, 382, 304, 423, 303, 466, 302, 511, 301, 558, 300, 607, 299, 658, 298, 711, 297, 766, 296, 823, 295, 882, 294, 943, 293, 1006, 292,
1071, 328, 342, 381, 341, 422, 340, 465, 339, 510, 338, 557, 337, 606, 336, 657, 335, 710, 334, 765, 333, 822, 332, 881, 331, 942, 330, 1005, 329,
1070, 367, 380, 421, 379, 464, 378, 509, 377, 556, 376, 605, 375, 656, 374, 709, 373, 764, 372, 821, 371, 880, 370, 941, 369, 1004, 368,
1069, 408, 420, 463, 419, 508, 418, 555, 417, 604, 416, 655, 415, 708, 414, 763, 413, 820, 412, 879, 411, 940, 410, 1003, 409,
1068, 451, 462, 507, 461, 554, 460, 603, 459, 654, 458, 707, 457, 762, 456, 819, 455, 878, 454, 939, 453, 1002, 452,
1067, 496, 506, 553, 505, 602, 504, 653, 503, 706, 502, 761, 501, 818, 500, 877, 499, 938, 498, 1001, 497,
1066, 543, 552, 601, 551, 652, 550, 705, 549, 760, 548, 817, 547, 876, 546, 937, 545, 1000, 544,
1065, 592, 600, 651, 599, 704, 598, 759, 597, 816, 596, 875, 595, 936, 594, 999, 593,
1064, 643, 650, 703, 649, 758, 648, 815, 647, 874, 646, 935, 645, 998, 644,
1063, 696, 702, 757, 701, 814, 700, 873, 699, 934, 698, 997, 697,
1062, 751, 756, 813, 755, 872, 754, 933, 753, 996, 752,
1061, 808, 812, 871, 811, 932, 810, 995, 809,
1060, 867, 870, 931, 869, 994, 868,
1059, 928, 930, 993, 929,
1058, 991, 992,
1057, 1056,
1123, 191, ....
```

Once the data arranged like that, the patterns seem quite predictable. However, every other block (e.g. the penultimate one, starting with $211=a_{113}$) has "paragraphs" of lengths $2$ or $3$, except possibly the first one. Now it can be seen by construction that from one block to the (over-)next one, the 2-3-sequence of the block is generated by the preceding one in a similar way as the "rabbit sequence", a.k.a. the "Fibonacci word" https://oeis.org/A005614, by the laws (essentially) $3\to22, 2\to323$ plus boundary conditions that are much harder to predict...

So this partly explains the self-similarity. But:

How can one possibly prove that the asymptotic ratio is $3+2\sqrt2$?

Each block consists of a bunch of "horizontal" and a bunch of "vertical" monotonous subsequences of decreasing lenghts. For every other block, those come in roughly "L-shaped" pairs. The numbers of "L-shapes" per block are clearly distinguishible, e.g. $8$ of them in the range of $n=49,\dots,112$ (starting after $a_{48}=190$) and $19$ for $n=270,\dots,630$ (starting after $a_{269}=1075$, the beginning of the last block). Those numbers $(c_j)=3,8,19,46,\dots$ seem to form the Fibonacci type sequence https://oeis.org/A078343 with $$c_j = \frac14 \Bigl[(3 \sqrt{2} - 2) (1 + \sqrt{2})^j - (3 \sqrt{2}+2) (1 - \sqrt{2})^j\Bigr],$$ which is another indication in favor of the conjectured ratio, but I am not sure whether the recursion $c_j=2c_{j-1}+c_{j-2}$ can be shown by induction.

You can also have a look at https://codegolf.stackexchange.com/a/145234/14614, which displays the differences $a_{n+1}-a_n$, and at the image of the inverse map $a_n\mapsto n$ quoted below from one of the comments. (Note that the isolated point at $a_n=191$ corresponds to $n=632$, which is just where my above table stops.) Both show a lot of beauty, but they also show that the self-similarity is somewhat less strict than for the fractal sequences mentioned here.