The best five examples is definitely a very vague thing. They could be examples of all kinds of things - the best five examples to show off what the games in the FF series are like would have to be varied, while the best five examples of the quality the series can achieve would have to be my favourite five. I'm going to go with the former.

I, V, VII, XI, XII.

By playing these games, I feel you're going a long way towards saying you have experienced a large amount of the gameplay mechanics, settings and atmospheres you can experience in the full series.

Note that I've not yet completed III, IV or IX. Was rough to not put in X, but I just didn't feel it was different enough from the first three games, at least not as much as XI & XII.