We describe approximation algorithms for MAX SAT with performance ratios arbitrarily close to 1, in particular, when performance ratios exceed the limits of polynomial-time approximation. Namely, given a polynomial-time α-approximation algorithm , we construct an -approximation algorithm . The algorithm runs in time of the order ck, where k is the number of clauses in the input formula and c is a constant depending on α. Thus we estimate the cost of improving a performance ratio. Similar constructions for MAX 2SAT (...) and MAX 3SAT are also described. Taking known algorithms as , we obtain particular upper bounds on the running time of. (shrink)