In adaptive control, it is often useful to distinguish between transient and asymptotic performance. In this paper, we formulate a notion of asymptotic performance for an H∞ adaptive control problem, and consider the problem in the simple case where the unknown plant is one of a finite number of known, possible models. We consider two plant cases: a) the plants are simply static nonlinear functions, and b) the plants are linear and time-invariant. Our main result is that, for both cases, the optimal asymptotic H∞ performance is no better than the optimal performance in the transient phase. We conclude that increased input-output data does not improve the achievable H∞ performance. Instead, parametric uncertainty results in a persistent performance degradation, and this uncertainty cannot be resolved, even with infinite data.