It is well known that estimating the expectation of any given bounded random variable with values in $[-B, B]$ has a sample complexity of $\mathrm{O}(B^2/ε^2)$ that is independent of the underlying probability measure. We show that this property can no longer hold when evaluating the worst-case expectation of the random variable, where the probability measures defining the expectation belong to a $φ$-divergence ball centered at some nominal measure $P$. Specifically, the sample complexity and its dependence on the nominal measure can be completely characterized by the growth of the divergence function. When the divergence function $φ$ exhibits superlinear growth, a $P$-independent sample complexity can be obtained for sample average approximation, which depends only on the growth of $φ$, the radius of the divergence ball, and the target precision. We also provide sample complexity lower bounds and demonstrate the optimality of the obtained bounds for commonly used $φ$-divergences. On the other hand, when superlinear growth does not hold for $φ$, we show that for any estimation method, evaluating the worst-case expectation has a $P$-dependent sample complexity lower bound that can be made arbitrarily large by changing $P$.