The next time there’s a SDMB debate about whether or not there can be confusion about the word “literally”, and whether people can always tell what it’s supposed to mean in a specific context … well, this thread could be Exhibit A.
Actually, Stephen King didn’t like him in the role. He wanted someone who wouldn’t telegraph the fact that the character had the potential for going crazy. He wanted someone who looked like he was so steady, he’d never go nuts. King’s choice for the role was Michael Moriarty.
This one is true; apparently he was Margaret Mitchell’s choice.
I’ll do you one better, and say that no one but Chaney could have given the performances Chaney gave, and if anyone mentioned in the thread is a candidate for the laurels, it’s him.
I wonder about the role of Helen Keller in The Miracle Worker. Patty Duke was amazing in this, and I say this as someone who knows several Deaf-blind people, including some when they were children. The movie has been remade several times, and I’ve seen them all, and I have personally seen four different stage productions, two amateur, one professional, and one semi-professional. The other movies were embarrassing. The stage versions I saw were better, but no one came close to Duke’s performance. She’s really under-rated, I think because she has a tendency to choose bad projects that don’t really challenge her. I never could figure out what was up with that.
It’s also hard to imagine anyone but Irene Dunne as Lucy Warriner, but that’s probably just me.
Lillian Gish and Bette Davis played aging sisters in The Whales of August. Gish and Davis were pretty old when they appeared in what turned out to be the last film for both of them. I think it would be hard to find equally experienced actresses who were as old, and still able to work, and able to do that film.
Similarly, Gish had a few roles where it’s hard to imagine someone else, because she was in fantastic shape when she was late middle aged and elderly. I remember one scene in a piece for The Alfred Hitchcock Hour where she has to run a distance on a backlot (outside, rough terrain), and it’s a medium shot, so it’s clearly her doing it. She was in her mid-sixties at the time, and isn’t even winded when it’s over. She also knew how to fire a rifle, and so could look convincing when threatening people off her land or away from her family in a couple of different movies. She was quite petite, but she’d lift a large hunting rifle like it was nothing.
So, I suppose there’s no role that really, literally could have been played by only one person, but there are some roles where the finished product would have been substantially different, and likely not very successful without that person in the role.
Gregory Peck as Atticus Finch.
I win. There is no other actor who could have given the role the gravitas and respect it needed.
Jimmy Stewart could’ve handled it if Henry Fonda was busy.
Exactly. People are not even comprehending the OP here. What a surprise.
Tattoo/ Nick-Nack.
Mini-Me.
Most roles can only be played by one person. You know, at a time.
Peter Sellers as Lionel Mandrake/President Muffley/Dr. Strangelove in the titular movie? Or are the multiple roles pushing the boundaries of the OP? 
For those claiming we didn’t read/understand the OP, sure… but then, apparently the OP didn’t read or understand the OP either, for while the two roles mentioned in the OP might be very well-suited to those actors, other actors could have handled them just fine, with little difference in the end result. 
This reminded me of Billy Bob Thornton, who created the character of Karl Childers while mugging in a mirror, then wrote the short film Some Folks Call It a Sling Blade for the character (George Hickenlooper directing), then wrote and directed the movie Sling Blade to flesh out the short film. Thornton was nominated for Best Actor for the role, and won the Best Adapted Screenplay Oscar.
Well, that’s certainly arguable. But certainly both examples in the OP are clearly different from Indiana-Jones-is-Iconically-Harrison-Ford.
And in fact a few of the examples in the thread DO fit what I was looking for, such as Audie Murphy playing Audie Murphy. Presumably they just wouldn’t have made that movie if he’d refused to be in it. Which is the point I was trying to make about Hugh Jackman… a big budget Hollywood adaptation of something like Les Miz is only going to get financed with a Hollywood A-lister as the star, and (as far as I know) the only Hollywood A-lister who has the appropriate Broadway singing background is Hugh Jackman.
Well, I thought Christopher “Saruman” Lee was Tolkein personal choice for Gandalf. I think he would have done a great job, but McKellen was simply wonderful.
What a cringe-worthy list, although I think Bridges, by far, was the least worst of the bunch. Would have been weird seeing him with short hair, though.
Robett Downey Jr., Tony Stark
Rex Harrison, Henry Higgins
Yul Brynner, King Mongkut of Siam
Benedict Cumberbatch, modern-day Sherlock Holmes
Other actors can and have played those roles, but their audiences were shortchanged.
Pauline Collins as Sarah in Upstairs, Downstairs. I realize that Eileen Atkins was originally slated for the role, but it was also originally going to be a comedy. I think someone could have done the role, but anyone else would have done it substantially differently, and it wouldn’t have been as satisfying. The character was disliked by the other downstairs staff, but immensely liked by the audience, and that took skill to pull off. I think if someone else had played it, either the character would have been disliked by the audience, or the things that rubbed the staff wrong would have been muted in order to make the character more likable.
Also, Pauline Collins real-life pregnancy was written into the script, and it was a very powerful storyline not just for her, but for the other characters directly involved, and had some far-reaching implications for one of them. That would not have happened with a different actress in the role.
Jennifer Saunders as Edina Monsoon. She created the character for herself, and did a lot of the writing. It actually grew out of a French & Saunders sketch (which was on youtube at one point). When an actor writes and creates a character as they go along, I think it’s fair to say that “literally” no one else could have done it, although maybe that twists the meaning of “literally” just a bit, but it’s still legitimate use of it.
Any role with a substantial amount of ad-libbing. I didn’t care for the show, but I understand that Redd Foxx largely ad-libbed the Fred Sanford role, so it’s probably fair to say that “literally,” no one else could have done it.
Jean-Pierre Leaud in Francois Truffaut’s The 400 Blows for a similar reason-- the script was very loose, and the exact wording of many lines was improvised. Almost all the scenes used were first takes.
Sure, all those projects might have been made with different actors, but would be so substantially different as to be different things, with a different emotional impact (and degree of success).
Short, BLACK hair with a curlicue in front? Possibly; I’d have to see the dailies. 
Yeah, you’re right about the acting credentials. I should have added a 
to my post.
Another shot at the OP:
Peter Sellers was mentioned for his roles in Dr. Strangelove and Hollywood would be hard pressed to find someone to fill all those shoes. But his Inspector Cleusseau (sp) character is undeniably all his. No one can touch him when it comes to the bumbling detective. Alan Arkin and Steve Martin both tried but they were no match for Sellers.
I haven’t seen anyone suggest Brando as Don Corleone or Al Pacino as Michael. (I’ve been skimming, so I could have missed it). I suppose it would have been possible to plug other actors into these parts, but the movie would never have been the blockbuster and historic classic that it has become. Witness The Godfather Part III, in which certain key actors (i.e., Robert Duvall and Winona Ryder) were either written out or replaced. The film, while a good mob movie, never attained the classic status of the first two.
So? Do you think they could have picked one of those quirky character actors to star in Being John Malkovich? I think they could have made a movie with a similar story but that’s not the same thing at all.
Had the OP not said literally I would vote for this one.
At that, Tarantino had an incredibly hard time finding someone who could earn an Oscar as Hans Landa: he needs to be an amiably sinister interrogator who can smoothly chat with German officers and French civilians and American troops and various Italians in Nazi-occupied France, or else the whole story falls apart.
But Hollywood has a long history of putting A-list starts who couldn’t sing very well in singing roles from Broadway adaptations - Marlon Brando in Guys and Dolls, Pierce Brosnan in Mama Mia, and heck, Russell Crowe in Les Mis. They can make do with movie magic. If they wanted someone else in that role the movie would have been made.