Optimal greedy algorithms for NP-hard problems
The method of conditional expectations (for derandomizing the "random assignment" algorithms for Max-Cut and Max-SAT) can be viewed as a greedy strategy: for $i=1,\ldots,n$, you pick the value of a variable $x_i$ such that the expected number of constraints satisfied in the resulting reduced instance exceeds the expected number of constraints satisfied in the current instance. (In fact, the greedy algorithm for $1/2$-approximating Max-Cut is the same as the "method of conditional expectations" algorithm for $1/2$-approximating Max-Cut.)

Since the method also works for Max-E3-SAT and achieves a $7/8$-approximation, this is an example of a greedy algorithm which is an optimal approximation unless $P=NP$ (cf. Hastad and Moshkovitz-Raz's inapproximability results for Max-E3-SAT).