On Azimuth, on May 6, 2016, Joseph van Name wrote:
The largest classical Laver table computed is actually $A_{48}$. The 48th table was computed by Dougherty and the algorithm was originally described in Dougherty's paper here. With today's technology I could imagine that one could compute $A_{96}$ if one has access to a sufficiently powerful computer.
One can compute the classical Laver tables up to the 48th table on your computer here at my website.