From a strict gaming point of view, I do think it is important that the protagonist has the chance to tangle with the villain multiple times throughout the story. If you only meet a villain once, for the final boss battle, who even cares? They should grow and progress as you do, to demonstrate that they aren't static creatures.

Let me use Ghirahim from Legend of Zelda: Skyward Sword as my example. You hate him plenty when you first see him, wanting to kill Zelda and being all bad-touch to Link, but he is an utter cliche - the 'effeminate as evil' trope and just generically full of murder. But you meet back up with him a few times, and he is changing. He is truly frustrated that he can't beat Link and can't capture Zelda, and increasingly unhinged by being a failure to his master. By the very end you are up against his true, fully powered form and he drops the floppish bullshit and is all business and badass. This progression is deeply satisfying because you, the hero, forced the villain to react to the threat you pose, and so you really feel like the villain is throwing the gloves off for you.

Contrast this against this example: Zant, in Legend of Zelda: Twilight Princess who manages to get LESS threatening each time you encounter him, making players feel like tools (or at least, I felt like a tool).

There are other ways you can follow the 'progression' of the villain in a game besides repeat boss battles. Kingdom Hearts has you collecting journal entries, The Last Story has the villain as an ally to your party for a time, Bioshock has you engaging in dialogue with them while you do other things.

Bravely Default is another excellent example of villain progression, though more in a boss rush format. If you want, you can relight all of the sub-bosses about 4 times, and each time you discover new and often touching facets about their characters. The first time you fought them, they seemed to you to be obviously evil, but in subsequent battles you discover who and what they really fought for, and you develop stronger sympathy and interest in (most) of them.