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 redcued forms and total count of fomrs, 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:
jagy@phobeusjunior:~$
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
d red red/(sqrt(d) log(d))
jagy@phobeusjunior:~$
