We know :
| Adj(A) | = |A|^{n-1}
Now extending this result we can solve as :
Adj(A) . Adj(Adj(A)) = |Adj(A)| . I where I is identity matrix
==> | Adj(A) . Adj(Adj(A)) | = |Adj(A)|^{n}
==> | Adj(Adj(A)) | = |Adj(A)|^{n-1} = |A|^{(n-1)^2}
==> | Adj(Adj(A)) * Adj( Adj(Adj(A)))| = | Adj(Adj(A)) |^{n}
==> | Adj( Adj(Adj(A))) | = | Adj(Adj(A)) |^{n-1}
==> | Adj( Adj(Adj(A))) | = | A |^{(n-1)^3}
Now n = 3 here and given |A| = 2
Therefore
| Adj( Adj(Adj(A))) | = 2^{2^3}
==> | Adj( Adj(Adj(A))) | = 256
Hence 256 is the correct answer..