[Wikipedia lists two articles on maximum length fo][1]r 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.][2] 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:~$




[![enter image description here][3]][2]


  [1]: https://en.wikipedia.org/wiki/Periodic_continued_fraction#Length_of_the_repeating_block
  [2]: http://www.rzuser.uni-heidelberg.de/~hb3/publ/bf.pdf
  [3]: https://i.sstatic.net/3d6kp.jpg