Is it buggy for at least 2^(n)^2? It gives 4 for any n, but surely for example 2^^2 = 2^(2^2) != 4?

2^^2 != 2^(2^2). Instead, 2^^2 = 2^2.

This will make more sense if you look at how the inputs a,b,n in the toy (2,2,3) and (2,3,3) present differently.