= Solution
For independent <Bernoulli random variables>, the <log-likelihood> is
$$
\ell(\pi)=\sum_{i=1}^n\{y_i\log\pi_i+(1-y_i)\log(1-\pi_i)\}.
$$
Every <likelihood> factor is at most one, so $\ell\le0$. In the <statistical saturated model>, choose $\widehat\pi_i=y_i$ independently for each observation. The probability of each realised response is then exactly one. With the continuous convention $0\log0=0$, the <Bernoulli saturated log-likelihood> is therefore
$$
\boxed{\ell_{\mathrm{sat}}=0\text{ for every binary response vector}.}
$$
The inclusion of probabilities zero and one is important: if the parameter space required $0<\pi_i<1$, zero would be the supremum approached at the boundary, rather than an attained maximum.
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