The hard-hexagons entropy constant is algebraic (see below z number).
The value is :
1.3954859724793027352295006635668880689541037281446611908174721561357608803586
977746898378730852754279026689685607685657184842212457119511639349818266947083
252547173794947534862281229126187281554340126162747356973585709823756812898414
948800016934903723995652094568253572538633572005211925074739811015138086289661
268136787831885630404682747107477204686894756657580905530270066675404962427719
060854536142216836296933016900330937276956621269398726823104923047442882514781
702966107270054292812280795061336321550953581179745072336957434963259935073449
490894249329307540816210555328068610619705545037955077580725537613858033619505
210958967729699416630942601615566925218549336476968551824281894615092855649748
501359906929152571833851080212811049755339847366927914398892041851355831303575
673710465224807454744982583885183287167357146092090743402851746571565499082292
999884612996137479952358336507860770516087879631202738350102895965881076822440
14681214726789035888008851819053742866660552775722734105313225337

Taken from 
The Favorite mathematical constants of Steven Finch, Mathsoft Inc.

The constant is given by this (see z below)...

                                       124   1/3
                                a := - --- 11
                                       363

                                      2501   1/2
                                b := ----- 33
                                     11979

         /       31   1/3 // 2501   1/2    \1/3   / 2501   1/2    \1/3\\1/3
    c := |1/4 - --- 11    ||----- 33    + 1|    - |----- 33    - 1|   ||
         \      242       \\11979          /      \11979          /   //

                                         1/4   7/12
                                        3    11
 z1 := 3/44 ------------------------------------------------------------------
            /       31   1/3 // 2501   1/2    \1/3   / 2501   1/2    \1/3\\2/3
            |1/4 - --- 11    ||----- 33    + 1|    - |----- 33    - 1|   ||
            \      242       \\11979          /      \11979          /   //

                      1/3 1/2          1/3            1/3     2/3 1/2 1/2 2
    z2 := (1 - (1 - %1   )    + (2 + %1    + 2 (1 + %1    + %1   )   )   )

               31   1/3 // 2501   1/2    \1/3   / 2501   1/2    \1/3\
%1 :=   1/4 - --- 11    ||----- 33    + 1|    - |----- 33    - 1|   |
              242       \\11979          /      \11979          /   /

                       1/3 1/2          1/3            1/3     2/3 1/2 1/2 2
   z3 := (- 1 - (1 - %1   )    + (2 + %1    + 2 (1 + %1    + %1   )   )   )

               31   1/3 // 2501   1/2    \1/3   / 2501   1/2    \1/3\
%1 :=   1/4 - --- 11    ||----- 33    + 1|    - |----- 33    - 1|   |
              242       \\11979          /      \11979          /   /

                                  1/3 1/2
      z4 := 1/(1/33 (1089 + 372 11   )

             /    124   1/3     /    124   1/3    15376   2/3\1/2\1/2
           + |2 - --- 11    + 2 |1 - --- 11    + ------ 11   |   |   )^1/2
             \    363           \    363         131769      /   /

            1/4   7/12
 z := 3/44 3    11

                 1/3 1/2          1/3            1/3     2/3 1/2 1/2 2
     (1 - (1 - %1   )    + (2 + %1    + 2 (1 + %1    + %1   )   )   )

                   1/3 1/2          1/3            1/3     2/3 1/2 1/2 2   /
     (- 1 - (1 - %1   )    + (2 + %1    + 2 (1 + %1    + %1   )   )   )   /  (
                                                                         /

       2/3                     1/3 1/2
     %1    (1/33 (1089 + 372 11   )

        /    124   1/3     /    124   1/3    15376   2/3\1/2\1/2
      + |2 - --- 11    + 2 |1 - --- 11    + ------ 11   |   |   )^1/2)
        \    363           \    363         131769      /   /

               31   1/3 // 2501   1/2    \1/3   / 2501   1/2    \1/3\
%1 :=   1/4 - --- 11    ||----- 33    + 1|    - |----- 33    - 1|   |
              242       \\11979          /      \11979          /   /

> evalf(z);

       1.395485972479302735229500663566888068954103728144661190817472165