He was the first person I thought of before I even opened the thread.
Hey! That’s my YouTube channel! Thanks for the plug. I have fantasies of someday hitting the 10[sup]6[/sup] mark on that one. Certainly none of my others are ever going to come anywhere near that close. 
You’re welcome! 
For the record, I had no idea that video was on a channel belonging to a Doper.
Another nomination for Edward Norton in Kingdom of Heaven. In the few minutes he actually appeared onscreen, I could feel my inner self declaring that I would gladly fight to the death to serve such a man.
I also nominate Lon Chaney, Sr., for his appearance in the 1925 version of The Phantom of the Opera. He does more acting with his eyes alone than many a competent performer of any day and age would do with their entire body, let alone a speaking voice.
Tim Curry as Pennywise the Clown in Stephen King’s IT.
I’m going with Andy Serkis. It’s his performance, but not his face, you see on screen as Gollum. (& Kong & the ape in the Rise of the Apes) In the case of Gollum, his success in the role was pivotal to the success of the film. I had no idea how they were going to manage this crazy character onscreen, but they did it.
If we’re talking Star Wars characters, Anthony Daniels is the hands-down winner.
If we’re heading in that direction, Tim Curry as Darkness in Legend.
Sorry to hijack but your user name popping up in this thread instantly authenticates it. 
Michael Chiklis in as The Thing was easily the best part of the Fantastic Four movies, another great “eyes only” performance.
John Forsythe as Charlie on Charlie’s Angels.
We did get to see what Thing looked like beforehand, however, and he gets a few minutes as “himself” in part 2 as well.
And I’ll add William Daniels from St. Elsewhere, playing the voice of KITT in Knightrider (he asked not to be credited according to the wiki).
What about the Addams Family’s Cousin ITT?
Odd example - Gary Oldman’s turn as the sinister Mason Verger from Hannibal. We see a face, but it’s not Gary Oldman’s face. Or Verger’s original face; it’s whatever was left after he cut it off. The more I think about that film the more convinced I am that I dreamed it.
I’ll throw in a Bill Nighy as Davy Jones in POTC 2 & 3.
Clayton Moore as The Lone Ranger.
I saw what you did there.
William S. Burroughs as the Clark Nova typewriter in Naked Lunch.