Let r*(n,m,f) denote the best approximation to a function f by a rational function which is the quotient of a polynomial of degree n by a polynomial of degree m. The following results are typical. r*(n,m,f) is a polynomial for all n and m if and only if f is a constant. r*(n,n,f) is a polynomial for all n if and only if f is a constant plus a multiple of a Tchebycheff polynomial. For any c 1, there exist continuous nonpolynomial functions f such that, for all n, r*(cn,n,f) is a polynomial. 20 pp.