Wikipedia lists two articles on maximum length for the continued fraction of $\sqrt n,$ Hickerson 1973 and Cohn 1977. There is a mess in the references, Cohn is not visible, just the link.
Hmmm. This guy does a review of one of the (Russian) references, http://www.jpr2718.org/podstotalc.pdf
I did my own modelling for two items, longest Gauss-Lagrange cycles of reduced forms, and largest number of reduced forms for a discriminant (positive but not a square). Reduced forms are $\langle a,b,c\rangle$ such that $\gcd(a,b,c) = 1, \; $ $d = b^2 - 4ac, \; \; $ $ac < 0, \; b > |a+c|.$ This equivalent version of reduction is in Franz's book, Theorem 1.36, formula 1.34. It is page numbered 37, pdf page 43.
Details: the longest individual cycles were all prime discriminant, class number one. The largest form count was sometimes composite discriminant, class number not one.
Hickerson and Cohn say that the period length for $\sqrt d$ is below $\sqrt d \log d.$ My own computations say this for both cycles of arbitrary reduced forms and total count of forms, where the latter has slightly larger implied constant (maybe over Cohn's $\frac{7}{2 \pi^2}.$ So the questions are, is $C_j \sqrt d \log d$ a provable upper bound for my two related problems?
Best length of individual cycle:
5 1 1 -1 2 3.598813 0.5557389
17 1 3 -2 6 11.68164 0.5136266
41 1 5 -4 10 23.77846 0.4205486
73 1 7 -6 18 36.6577 0.4910291
193 1 13 -6 30 73.11163 0.4103314
241 1 15 -4 38 85.14695 0.4462873
337 1 17 -12 42 106.8425 0.3931019
409 1 19 -12 54 121.6199 0.4440064
601 1 23 -18 66 156.8635 0.420748
769 1 27 -10 70 184.274 0.3798691
1033 1 31 -18 78 223.061 0.3496802
1201 1 33 -28 106 245.7386 0.4313526
1609 1 39 -22 118 296.1642 0.3984276
1801 1 41 -30 130 318.1208 0.4086498
2161 1 45 -34 146 356.939 0.4090335
2521 1 49 -30 170 393.2619 0.4322819
3361 1 57 -28 178 470.7496 0.3781203
3529 1 59 -12 198 485.2689 0.4080212
4201 1 63 -58 210 540.7576 0.3883441
4561 1 67 -18 214 569.0039 0.3760958
5209 1 71 -42 238 617.6703 0.3853188
5569 1 73 -60 258 643.6448 0.4008422
6841 1 81 -70 290 730.3893 0.3970485
7561 1 85 -84 306 776.5653 0.3940428
8089 1 89 -42 330 809.2934 0.4077631
9241 1 95 -54 346 877.8031 0.3941658
12049 1 109 -42 378 1031.46 0.3664707
12289 1 109 -102 390 1043.869 0.3736102
12601 1 111 -70 394 1059.851 0.3717503
13729 1 117 -10 426 1116.317 0.3816119
15649 1 125 -6 454 1208.197 0.3757665
16921 1 129 -70 474 1266.507 0.3742578
18481 1 135 -64 502 1335.59 0.3758639
19009 1 137 -60 522 1358.418 0.3842705
20161 1 141 -70 530 1407.329 0.3765999
21121 1 145 -24 542 1447.206 0.3745147
21961 1 147 -88 566 1481.483 0.3820495
24049 1 155 -6 578 1564.397 0.3694714
26041 1 161 -30 590 1640.741 0.3595937
26161 1 161 -60 602 1645.26 0.3658996
28081 1 167 -48 622 1716.434 0.3623793
28729 1 169 -42 630 1739.992 0.3620706
31249 1 175 -156 674 1829.564 0.3683938
33049 1 181 -72 702 1891.701 0.3710947
33289 1 181 -132 714 1899.877 0.3758138
38329 1 195 -76 722 2066.233 0.3494282
40609 1 201 -52 750 2138.444 0.3507222
43201 1 207 -88 766 2218.496 0.345279
43801 1 209 -30 794 2236.735 0.3549817
47041 1 215 -204 842 2333.464 0.3608369
47881 1 217 -198 862 2358.079 0.3655518
48049 1 219 -22 878 2362.98 0.3715648
49009 1 221 -42 886 2390.848 0.3705798
51769 1 227 -60 914 2469.714 0.3700834
53881 1 231 -130 966 2528.87 0.3819888
59929 1 243 -220 974 2693.068 0.3616693
61681 1 247 -168 1002 2739.307 0.365786
65521 1 255 -124 1006 2838.747 0.3543817
66361 1 257 -78 1022 2860.168 0.3573217
67369 1 259 -72 1042 2885.721 0.3610882
69001 1 261 -220 1074 2926.753 0.3669596
70849 1 265 -156 1086 2972.721 0.3653219
80809 1 283 -180 1142 3212.198 0.3555198
87481 1 295 -114 1242 3365.64 0.3690234
101641 1 317 -288 1246 3675.646 0.338988
101929 1 319 -42 1270 3681.754 0.3449443
102001 1 319 -60 1298 3683.279 0.3524034
==========================================
Best total count of reduced forms:
file named many_reduced_binary.txt
d red red/(sqrt(d) log(d))
5 2 0.555739 5 = 5
12 4 0.464686 12 = 2^2 * 3
17 6 0.513627 17 = 17
28 8 0.453711 28 = 2^2 * 7
41 10 0.420549 41 = 41
57 12 0.393129 57 = 3 * 19
73 18 0.491029 73 = 73
105 20 0.419385 105 = 3 * 5 * 7
145 28 0.467229 145 = 5 * 29
193 30 0.410331 193 = 193
217 32 0.403781 217 = 7 * 31
241 38 0.446287 241 = 241
265 40 0.440376 265 = 5 * 53
337 42 0.393102 337 = 337
385 44 0.376677 385 = 5 * 7 * 11
409 54 0.444006 409 = 409
481 56 0.413445 481 = 13 * 37
505 60 0.42894 505 = 5 * 101
601 66 0.420748 601 = 601
649 68 0.412209 649 = 11 * 59
721 72 0.407471 721 = 7 * 103
865 80 0.402217 865 = 5 * 173
889 84 0.414909 889 = 7 * 127
1009 90 0.409635 1009 = 1009
1081 92 0.400561 1081 = 23 * 47
1129 102 0.431871 1129 = 1129
1201 106 0.431353 1201 = 1201
1489 114 0.404377 1489 = 1489
1609 118 0.398428 1609 = 1609
1801 130 0.40865 1801 = 1801
1969 140 0.415943 1969 = 11 * 179
2161 146 0.409034 2161 = 2161
2521 170 0.432282 2521 = 2521
3241 180 0.391135 3241 = 7 * 463
3529 198 0.408021 3529 = 3529
3649 208 0.419803 3649 = 41 * 89
4201 210 0.388344 4201 = 4201
4321 216 0.392529 4321 = 29 * 149
4369 220 0.397072 4369 = 17 * 257
4729 230 0.395273 4729 = 4729
5209 238 0.385319 5209 = 5209
5401 240 0.379981 5401 = 11 * 491
5569 258 0.400842 5569 = 5569
6049 264 0.389817 6049 = 23 * 263
6169 272 0.396809 6169 = 31 * 199
6769 276 0.380341 6769 = 7 * 967
6841 290 0.397049 6841 = 6841
7561 306 0.394043 7561 = 7561
8089 330 0.407763 8089 = 8089
9241 346 0.394166 9241 = 9241
9529 352 0.393572 9529 = 13 * 733
10921 380 0.391059 10921 = 67 * 163
12289 390 0.37361 12289 = 12289
12601 394 0.37175 12601 = 12601
12961 404 0.374736 12961 = 13 * 997
13729 426 0.381612 13729 = 13729
14281 434 0.37962 14281 = 14281
14569 448 0.387165 14569 = 17 * 857
15409 472 0.394326 15409 = 19 * 811
15961 480 0.392582 15961 = 11 * 1451
17329 492 0.382933 17329 = 13 * 31 * 43
18001 516 0.392515 18001 = 47 * 383
19009 522 0.38427 19009 = 19009
20161 530 0.3766 20161 = 20161
20689 532 0.372195 20689 = 17 * 1217
21121 542 0.374515 21121 = 21121
21961 566 0.38205 21961 = 21961
23689 574 0.370245 23689 = 23689
23809 576 0.370412 23809 = 29 * 821
23881 584 0.374878 23881 = 11 * 13 * 167
25249 596 0.370028 25249 = 7 * 3607
26161 602 0.3659 26161 = 26161
27049 616 0.367007 27049 = 11 * 2459
28081 622 0.362379 28081 = 28081
28681 660 0.379691 28681 = 23 * 29 * 43
31201 680 0.372014 31201 = 41 * 761
33049 702 0.371095 33049 = 33049
33289 714 0.375814 33289 = 33289
37129 752 0.3709 37129 = 107 * 347
37801 756 0.368915 37801 = 103 * 367
40441 768 0.360025 40441 = 37 * 1093
40681 776 0.362499 40681 = 17 * 2393
43801 794 0.354982 43801 = 43801
43849 808 0.361006 43849 = 13 * 3373
44209 812 0.361037 44209 = 11 * 4019
44641 826 0.365148 44641 = 44641
45049 848 0.372856 45049 = 19 * 2371
46561 852 0.36735 46561 = 101 * 461
47881 862 0.365552 47881 = 47881
48049 878 0.371565 48049 = 48049
49009 886 0.37058 49009 = 49009
50521 912 0.374649 50521 = 19 * 2659
51769 914 0.370083 51769 = 51769
53881 966 0.381989 53881 = 53881
58969 984 0.368886 58969 = 109 * 541
61681 1002 0.365786 61681 = 61681
63361 1022 0.367213 63361 = 63361
65209 1040 0.367392 65209 = 61 * 1069
65641 1052 0.370186 65641 = 41 * 1601
69001 1074 0.36696 69001 = 69001
70849 1086 0.365322 70849 = 70849
74281 1100 0.359858 74281 = 59 * 1259
74881 1128 0.367273 74881 = 103 * 727
77401 1144 0.365291 77401 = 17 * 29 * 157
81481 1180 0.365564 81481 = 17 * 4793
84529 1200 0.363814 84529 = 137 * 617
85801 1212 0.364239 85801 = 239 * 359
86641 1252 0.374111 86641 = 23 * 3767
92569 1286 0.369611 92569 = 92569
95209 1296 0.366384 95209 = 19 * 5011
100321 1320 0.361886 100321 = 13 * 7717
105169 1328 0.354137 105169 = 251 * 419
105361 1330 0.354291 105361 = 105361
109729 1360 0.353756 109729 = 197 * 557
111721 1378 0.354679 111721 = 111721
114409 1396 0.354341 114409 = 191 * 599
117049 1496 0.374683 117049 = 67 * 1747
127009 1516 0.361968 127009 = 107 * 1187
130729 1558 0.365766 130729 = 130729
138121 1584 0.360101 138121 = 37 * 3733
144169 1610 0.35696 144169 = 144169
153169 1672 0.357826 153169 = 89 * 1721
155401 1676 0.355666 155401 = 19 * 8179
157249 1680 0.354064 157249 = 67 * 2347
157921 1736 0.364956 157921 = 79 * 1999
173209 1814 0.361346 173209 = 173209
180049 1852 0.360682 180049 = 401 * 449
185641 1902 0.363879 185641 = 185641
197521 1910 0.352448 197521 = 197521
204649 1916 0.346336 204649 = 19 * 10771
204961 1948 0.351809 204961 = 97 * 2113
210961 1978 0.351281 210961 = 210961
220081 1992 0.345168 220081 = 29 * 7589
223609 2104 0.361221 223609 = 311 * 719
242449 2114 0.346277 242449 = 242449
251329 2144 0.343934 251329 = 13 * 19333
259009 2170 0.342077 259009 = 259009
261361 2200 0.344992 261361 = 31 * 8431
266809 2216 0.343367 266809 = 367 * 727
268129 2272 0.351037 268129 = 73 * 3673
279889 2316 0.349039 279889 = 19 * 14731
287401 2396 0.355594 287401 = 31 * 73 * 127
295201 2398 0.350411 295201 = 295201
306289 2412 0.345008 306289 = 17 * 43 * 419
314161 2446 0.344768 314161 = 314161
315121 2452 0.345004 315121 = 181 * 1741
316369 2476 0.347584 316369 = 19 * 16651
320401 2534 0.353128 320401 = 320401
328441 2600 0.357165 328441 = 53 * 6197
349801 2602 0.344645 349801 = 349801
357361 2656 0.347474 357361 = 191 * 1871
360721 2664 0.34664 360721 = 137 * 2633
367249 2692 0.346671 367249 = 181 * 2029
373129 2700 0.344523 373129 = 23 * 16223
383881 2716 0.340922 383881 = 103 * 3727
389041 2730 0.340046 389041 = 389041
393121 2782 0.344441 393121 = 393121
394969 2866 0.353881 394969 = 394969
415801 2878 0.344971 415801 = 415801
416329 2936 0.351665 416329 = 263 * 1583
425569 2952 0.34913 425569 = 23 * 18503
457081 2996 0.340027 457081 = 83 * 5507
463849 2998 0.337382 463849 = 463849
465481 3008 0.337823 465481 = 19 * 24499
469249 3032 0.338939 469249 = 11 * 29 * 1471
469561 3038 0.339479 469561 = 469561
483289 3154 0.346635 483289 = 483289
503281 3212 0.344859 503281 = 463 * 1087
511681 3244 0.344989 511681 = 23 * 22247
515761 3330 0.352519 515761 = 515761
528889 3336 0.348078 528889 = 167 * 3167
540409 3348 0.345023 540409 = 439 * 1231
552841 3386 0.3444 552841 = 552841
563161 3432 0.345382 563161 = 233 * 2417
579049 3448 0.341481 579049 = 31 * 18679
600601 3542 0.343493 600601 = 600601
601129 3560 0.345064 601129 = 509 * 1181
605641 3570 0.344549 605641 = 605641
628489 3668 0.346549 628489 = 41 * 15329
663961 3706 0.339262 663961 = 663961
677161 3728 0.337438 677161 = 17 * 61 * 653
685441 3812 0.342641 685441 = 31 * 22111
691681 3846 0.343902 691681 = 691681
702529 3866 0.342615 702529 = 702529
722929 3920 0.341737 722929 = 79 * 9151
723529 3950 0.344188 723529 = 723529
766921 4004 0.337423 766921 = 17 * 197 * 229
772489 4024 0.337704 772489 = 31 * 24919
786241 4134 0.343441 786241 = 786241
799681 4136 0.340283 799681 = 37 * 21613
819289 4174 0.338671 819289 = 819289
825841 4256 0.343751 825841 = 379 * 2179
878641 4398 0.342822 878641 = 878641
914281 4416 0.336472 914281 = 337 * 2713
924169 4444 0.336525 924169 = 881 * 1049
928489 4528 0.341971 928489 = 17 * 54617
969049 4590 0.338269 969049 = 969049
979969 4698 0.344014 979969 = 979969
1007161 4706 0.339243 1007161 = 1007161
1057081 4792 0.336012 1057081 = 157 * 6733
1083289 4966 0.343368 1083289 = 1083289
1162561 4988 0.33124 1162561 = 103 * 11287
1171921 5070 0.335145 1171921 = 1171921
1195489 5118 0.33449 1195489 = 1195489
1201561 5300 0.345384 1201561 = 541 * 2221
1264561 5322 0.336838 1264561 = 1264561
1267249 5424 0.342878 1267249 = 31 * 40879
1354369 5468 0.332783 1354369 = 73 * 18553
1354609 5480 0.333479 1354609 = 71 * 19079
1368481 5532 0.334692 1368481 = 29 * 47189
1379569 5576 0.335803 1379569 = 43 * 32083
1391329 5624 0.337057 1391329 = 83 * 16763
1433041 5826 0.343328 1433041 = 1433041
1466089 5856 0.340636 1466089 = 23 * 63743
1538161 5864 0.331892 1538161 = 1063 * 1447
1552849 5900 0.332125 1552849 = 229 * 6781
1577689 5910 0.329692 1577689 = 1577689
1609609 5920 0.3265 1609609 = 23 * 47 * 1489
1627921 6120 0.335362 1627921 = 571 * 2851
1642369 6132 0.334331 1642369 = 433 * 3793
1672609 6158 0.332276 1672609 = 1672609
1699681 6210 0.332031 1699681 = 1699681
1701001 6232 0.33306 1701001 = 31 * 37 * 1483
1719481 6240 0.331441 1719481 = 19 * 90499
1720489 6304 0.334729 1720489 = 97 * 17737
1739929 6312 0.333015 1739929 = 83 * 20963
1785169 6444 0.335045 1785169 = 149 * 11981
1848289 6534 0.333069 1848289 = 1848289
1851529 6604 0.336302 1851529 = 167 * 11087
1906129 6768 0.338999 1906129 = 37 * 51517
2031649 6964 0.336386 2031649 = 53 * 38333
2144041 7062 0.330832 2144041 = 2144041
2159161 7128 0.332592 2159161 = 1117 * 1933
2213929 7156 0.329177 2213929 = 433 * 5113
2222089 7226 0.331703 2222089 = 2222089
2235601 7236 0.331019 2235601 = 569 * 3929
2293729 7402 0.333709 2293729 = 2293729
2356201 7408 0.328919 2356201 = 479 * 4919
2413321 7604 0.333058 2413321 = 23 * 317 * 331
2490049 7620 0.327878 2490049 = 23 * 108263
2519449 7636 0.326384 2519449 = 73 * 34513
2545321 7772 0.330275 2545321 = 41 * 62081
2602681 7886 0.330907 2602681 = 2602681
2636209 7892 0.328761 2636209 = 31 * 277 * 307
2675401 7952 0.328497 2675401 = 281 * 9521
2770681 8060 0.326412 2770681 = 53 * 61 * 857
2784961 8234 0.332487 2784961 = 2784961
2922529 8254 0.324302 2922529 = 2922529
2938849 8348 0.326961 2938849 = 59 * 49811
2959441 8468 0.33035 2959441 = 383 * 7727
2985889 8548 0.331793 2985889 = 31 * 61 * 1579
3100969 8596 0.326578 3100969 = 1319 * 2351
3147769 8604 0.324118 3147769 = 1123 * 2803
3197041 8664 0.323518 3197041 = 397 * 8053
3199681 8708 0.325009 3199681 = 41 * 78041
3206641 8788 0.327591 3206641 = 643 * 4987
3267289 8810 0.324942 3267289 = 3267289
3335929 9140 0.333166 3335929 = 101 * 33029
3395401 9220 0.332734 3395401 = 709 * 4789
3528001 9242 0.326369 3528001 = 3528001
3581449 9356 0.327593 3581449 = 89 * 40241
3674449 9412 0.324805 3674449 = 227 * 16187
3712921 9628 0.330306 3712921 = 43 * 79 * 1093
3818929 9818 0.331499 3818929 = 3818929
3864169 9860 0.330706 3864169 = 37 * 181 * 577
4064449 9904 0.322818 4064449 = 883 * 4603
4217929 9910 0.316312 4217929 = 4217929
4269841 10050 0.318569 4269841 = 4269841
4270561 10092 0.31987 4270561 = 47 * 90863
4275049 10150 0.321518 4275049 = 4275049
4280329 10378 0.32851 4280329 = 4280329
4499881 10420 0.320642 4499881 = 23 * 179 * 1093
4586761 10660 0.324501 4586761 = 563 * 8147
4701841 10770 0.32329 4701841 = 4701841
4708729 10830 0.324822 4708729 = 4708729
4724281 10932 0.327271 4724281 = 43 * 181 * 607
4861081 11056 0.325688 4861081 = 859 * 5659
4962049 11062 0.322102 4962049 = 4962049
5013649 11082 0.320805 5013649 = 5013649
5048761 11220 0.323522 5048761 = 37 * 136453
5098129 11354 0.325591 5098129 = 5098129
5343529 11410 0.318626 5343529 = 5343529
5368609 11438 0.318564 5368609 = 5368609
5390449 11566 0.321392 5390449 = 5390449
5505361 11602 0.318577 5505361 = 5505361
5542681 11684 0.319607 5542681 = 293 * 18917
5587201 11744 0.319801 5587201 = 73 * 76537
5634841 11984 0.324777 5634841 = 37 * 152293
5730169 12104 0.324938 5730169 = 233 * 24593
5815681 12308 0.327664 5815681 = 71 * 101 * 811
6016921 12418 0.324309 6016921 = 6016921
6080881 12430 0.322692 6080881 = 6080881
6104641 12586 0.326024 6104641 = 6104641
6195289 12648 0.324918 6195289 = 67 * 92467
6406009 12912 0.325503 6406009 = 199 * 32191
6589969 13026 0.323177 6589969 = 6589969
6763849 13046 0.318957 6763849 = 6763849
6783529 13194 0.322047 6783529 = 6783529
6868801 13434 0.325605 6868801 = 6868801
6969169 13504 0.324637 6969169 = 967 * 7207
7209409 13856 0.326799 7209409 = 127 * 56767
7478689 14180 0.327604 7478689 = 43 * 173923
7623529 14206 0.324679 7623529 = 7623529
7880161 14332 0.321509 7880161 = 47 * 167663
7926361 14420 0.32242 7926361 = 359 * 22079
8092681 14484 0.320087 8092681 = 79 * 89 * 1151
8254801 14666 0.320511 8254801 = 8254801
8281729 14724 0.32119 8281729 = 47 * 176207
8465209 15118 0.325743 8465209 = 8465209
8895769 15350 0.321638 8895769 = 8895769
9033649 15706 0.326263 9033649 = 9033649
9257329 15854 0.324838 9257329 = 9257329
9850801 15906 0.314714 9850801 = 9850801
9853561 15992 0.316366 9853561 = 173 * 56957
10018849 16024 0.314049 10018849 = 47 * 59 * 3613
10058161 16340 0.319538 10058161 = 41 * 245321
10319761 16586 0.319702 10319761 = 10319761
=======================================================
