LAVER TABLES data base Laver Tables are left-self distributive algebraic structures (see question 8. What's the latest on Laver Tables?) or http://en.wikipedia.org/wiki/Laver_table 
I wonder if there is any available compilation of tables from A0 up to order A20 or more (the last one has 2^40 elements). The first ones (A0, A1, A2, A3, A4) are published in Dehornoy papers. I know that the  calculated upper limit is at least A28.
Thanks.
 A: Added 3/15/2016
At my website http://boolesrings.org/jvanname/lavertables, I have started writing a database on Laver tables where one can compute the Laver table operations and many other things about the Laver tables online. I should also mention that Randall Dougherty has computed the classical Laver tables up to $A_{48}$. With all this being said, in order to better understand Laver tables, one needs to compute many things such as congruences on subalgebras of Laver tables rather than simply computing larger and larger classical Laver tables. I therefore am developing computational resources on Laver tables that compute much more than simply the application operation on classical Laver tables. These more complex computations for me work well up to around $A_{8}$ to $A_{13}$ (depending on what we want to compute) but after this point, the calculations generally become too computationally intensive. In fact, I have worked on algebras that one can construct from the classical Laver tables that become far too computationally intensive even when working with $A_{4}$. The moral of this story is that much still remains to be understood even with small classical Laver tables which can be computed by hand at one setting while sitting on a train or waiting at the DMV.
Added 11/19/2015
This is not quite a database for the classical Laver tables, but this is the code in the language GAP that I wrote. The code generates all classical Laver tables up to $A_{20}$ on my computer before it runs out of memory. This code basically follows the double recursive construction of the classical Laver tables in the proof of Lemma 1.2 in Chapter 10 of Dehornoy's book Braids and Self-distributivity except for the fact that the periods are only calculated once since rows of the classical Laver tables are periodic. The code is not optimal since much larger classical Laver tables have been calculated.
limit:=20;
fastlavertable:=[];
for v in [1..limit] do
  fastlavertable[v]:=[];
  for i in [1..2^v] do 
    fastlavertable[v][i]:=[i+1]; 
  od;
  fastlavertable[v][2^v]:=[1..2^v];
  ll:=Reversed([1..2^v-1]);
  for i in ll do
    k:=1;
    while fastlavertable[v][i][k]<2^v do
      fastlavertable[v][i][k+1]:=
        fastlavertable[v][fastlavertable[v][i][k]]
          [RemInt(i,Length(fastlavertable[v][fastlavertable[v][i][k]]))+1];
      k:=k+1;
    od;
  od;
od;

lavertablefunction:=function(r,x,y)
return fastlavertable[r][x][RemInt(y-1,Length(fastlavertable[r][x]))+1];
end;

lavertablefunction(r,x,y); produces the output $x*y$ in the Laver table $A_{r}$.
fastlavertable[r][x]; produces one period the $x$-th row in $A_{r}$.
I let limit:=20 since I wanted to calculate the classical Laver tables up to $A_{20}$, but one can use any natural number instead.
