Researcher John Tromp recently finished computing the number of legal positions on a 18X18 go board. It took 9 months of computations and 4 petabytes of disk I/O. He’s working on 19X19 now.
More information here: http://tromp.github.io/go/legal.html
For reference, here’s the number: 669723114288829212892740188841706543509937780640178732810318337696945624428547218105214326012774371397184848890970111836283470468812827907149926502347633, or ~6.6972e+152