These sequences are of course infinite, but for my convenience, I only plan to go up to about 1000 in each chain, and up to the self number closest to 200.

The initial terms of the following sequences correspond to the self numbers (Sloane's A003052).

1, 2, 4, 8, 16, 23, 28, 38, 49, 62, 70, 77, 91, 101, 103, 107, 115, 122, 127, 137, 148, 161, 169, 185, 199, 218, 229, 242, 250, 257, 271, 281, 292, 305, 313, 320, 325, 335, 346, 359, 376, 392, 406, 416, 427, 440, 448, 464, 478, 497, 517, 530, 538, 554, 568, 587, 607, 620, 628, 644, 658, 677, 697, 719, 736, 752, 766, 785, 805, 818, 835, 851, 865, 884, 904, 917, 934, 950, 964, 983, 1003

Chain of digit addition generators with initial term 1. After initial term, each term is equal to the previous term plus the sum of its digits.

Thanks to Alonso Delarte for this program. He writes that this program can easily be modified for any other digit addition generator chain, and in fact, I have used it for each of the following chains.

This sequence is Sloane's A004207.

3, 6, 12, 15, 21, 24, 30, 33, 39, 51, 57, 69, 84, 96, 111, 114, 120, 123, 129, 141, 147, 159, 174, 186, 201, 204, 210, 213, 219, 231, 237, 249, 264, 276, 291, 303, 309, 321, 327, 339, 354, 366, 381, 393, 408, 420, 426, 438, 453, 465, 480, 492, 507, 519, 534, 546, 561, 573, 588, 609, 624, 636, 651, 663, 678, 699, 723, 735, 750, 762, 777, 798, 822, 834, 849, 870, 885, 906, 921, 933, 948, 969, 993, 1014

Chain of digit addition generators with initial term 3. After initial term, each term is equal to the previous term plus the sum of its digits. A little reflection will show that every term of this sequence must be a multiple of 3.

Sloane's A016052.

5, 10, 11, 13, 17, 25, 32, 37, 47, 58, 71, 79, 95, 109, 119, 130, 134, 142, 149, 163, 173, 184, 197, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620, 628, 644, 658, 677, 697, 719, 736, 752, 766, 785, 805, 818, 835, 851, 865, 884, 904, 917, 934, 950, 964, 983, 1003

Chain of digit addition generators with initial term 5. After initial term, each term is equal to the previous term plus the sum of its digits.

Sloane's A007618.

7, 14, 19, 29, 40, 44, 52, 59, 73, 83, 94, 107

Chain of digit addition generators with initial term 7. After initial term, each term is equal to the previous term plus the sum of its digits. At 107, this sequence converges with DAGCH00001.

Sloane's A006507.

9, 18, 27, 36, 45, 54, 63, 72, 81, 90, 99, 117, 126, 135, 144, 153, 162, 171, 180, 189, 207, 216, 225, 234, 243, 252, 261, 270, 279, 297, 315, 324, 333, 342, 351, 360, 369, 387, 405, 414, 423, 432, 441, 450, 459, 477, 495, 513, 522, 531, 540, 549, 567, 585, 603, 612, 621, 630, 639, 657, 675, 693, 711, 720, 729, 747, 765, 783, 801, 810, 819, 837, 855, 873, 891, 909, 927, 945, 963, 981, 999, 1026

Chain of digit addition generators with initial term 9. After initial term, each term is equal to the previous term plus the sum of its digits. A little reflection will show that every term of this sequence must be a multiple of 9. As a matter of fact, you have to go the 12th term to verify that this sequence is in fact not equal to the sequence of multiples of 9.

Sloane's A016096.

20, 22, 26, 34, 41, 46, 56, 67, 80, 88, 104, 109

Chain of digit addition generators with initial term 20. After initial term, each term is equal to the previous term plus the sum of its digits. At 109, this sequence converges with DAGCH00005.

Sloane's A036227.

53, 61, 68, 82, 92, 103

Chain of digit addition generators with initial term 53. After initial term, each term is equal to the previous term plus the sum of its digits. At 103, this sequence converges with DAGCH00001.

64, 74, 85, 98, 115

Chain of digit addition generators with initial term 64. After initial term, each term is equal to the previous term plus the sum of its digits. At 115, this sequence converges with DAGCH00001.

75, 87, 102, 105, 111

Chain of digit addition generators with initial term 75. After initial term, each term is equal to the previous term plus the sum of its digits. At 111, this sequence converges with DAGCH00003.

86, 100, 101

Chain of digit addition generators with initial term 86. After initial term, each term is equal to the previous term plus the sum of its digits. At 101, this sequence converges with DAGCH00001.

97, 113, 118, 128, 139, 152, 160, 167, 181, 191, 202, 206, 214, 221, 226, 236, 247, 260, 268, 284, 298, 317, 328, 341, 349, 365, 379, 398, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620

Chain of digit addition generators with initial term 64. After initial term, each term is equal to the previous term plus the sum of its digits. At 115, this sequence converges with DAGCH00001.

108, 117

Chain of digit addition generators with initial term 108. After initial term, each term is equal to the previous term plus the sum of its digits. At 117, this sequence converges with DAGCH00009.

110, 112, 116, 124, 131, 136, 146, 157, 170, 178, 194, 208, 218

Chain of digit addition generators with initial term 110. After initial term, each term is equal to the previous term plus the sum of its digits. At 218, this sequence converges with DAGCH00001.

Sloane's A036230.

121, 125, 133, 140, 145, 155, 166, 179, 196, 212, 217, 227, 238, 251, 259, 275, 289, 308, 319, 332, 340, 347, 361, 371, 382, 395, 412, 419, 433, 443, 454, 467, 484, 500, 505, 515, 526, 539, 556, 572, 586, 605, 616, 629, 646, 662, 676, 695, 715, 728, 745, 761, 775, 794, 814, 827, 844, 860, 874, 893, 913, 926, 943, 959, 982, 1001

Chain of digit addition generators with initial term 121. After initial term, each term is equal to the previous term plus the sum of its digits.

Sloane's A036231.

132, 138, 150, 156, 168, 183, 195, 210

Chain of digit addition generators with initial term 132. After initial term, each term is equal to the previous term plus the sum of its digits. At 210, this sequence converges with DAGCH00003.

143, 151, 158, 172, 182, 193, 206

Chain of digit addition generators with initial term 143. After initial term, each term is equal to the previous term plus the sum of its digits. At 206, this sequence converges with DAGCH00097, which in turn converges with DAGCH00001 at 620.

154, 164, 175, 188, 205, 212

Chain of digit addition generators with initial term 154. After initial term, each term is equal to the previous term plus the sum of its digits. At 212, this sequence converges with DAGCH00121.

165, 177, 192, 204

Chain of digit addition generators with initial term 165. After initial term, each term is equal to the previous term plus the sum of its digits. At 204, this sequence converges with DAGCH00003.

176, 190, 200, 202

Chain of digit addition generators with initial term 176. After initial term, each term is equal to the previous term plus the sum of its digits. At 202, this sequence converges with DAGCH00097.

Coincidentally, the four terms given here appear in Sloane's A067943, a sequence involving repunits and modular arithmetic. The term after 202 in Sloane's A067943 is 205, differing from the next term 206 in this one which matches DAGCH00097.

187, 203, 208, 218

Chain of digit addition generators with initial term 187. After initial term, each term is equal to the previous term plus the sum of its digits. At 218, this sequence converges with DAGCH00001.

198, 216

Chain of digit addition generators with initial term 198. After initial term, each term is equal to the previous term plus the sum of its digits. At 216, this sequence converges with DAGCH00009.

Coincidentally, terms from 198 to 297 in this sequence are in Sloane's A045776, a(n + 1) is smallest multiple of (sum of digits of a(n)) which is > a(n). Given that these are both base-dependent sequences involving sums of digits, the coincidence is enticing.

209, 220, 224, 232, 239, 253, 263, 274, 287, 304, 311, 316, 326, 337, 350, 358, 374, 388, 407, 418, 431, 439, 455, 469, 488, 508, 521, 529, 545, 559, 578, 598, 620

Chain of digit addition generators with initial term 209. After initial term, each term is equal to the previous term plus the sum of its digits. At 620, this sequence converges with DAGCH00001.

211, 215, 223, 230, 235, 245, 256, 269, 286, 302, 307, 317, 328, 341, 349, 365, 379, 398, 418

Chain of digit addition generators with initial term 211. After initial term, each term is equal to the previous term plus the sum of its digits. At 418, this sequence converges with DAGCH00209.