Did Frodo lose a finger on Mount Doom?

Question

In the Return of the King film, Gollum bites the finger off Frodo. In later scenes, like the Coronation of Aragorn and Leaving the Shire, we see that Frodo has gotten the finger back.

How is this possible? How did Frodo get his finger back?

Answer 1

Yes, he lost his finger

In the film, Frodo loses the top part of his left index finger. The first image in your question proves this, and we can see Gollum discard it in another shot:

This is notably in contrast to the text, where Frodo loses the ring finger of his right hand (emphasis mine):

[Sam] sat up and then he saw that Frodo was lying beside him, and slept peacefully, one hand behind his head, and the other resting upon the coverlet. It was the right hand, and the third finger was missing.

^{Return of the King Book VI Chapter 4: "The Field of Cormallen"}

No, he never got it back

In later scenes, we either can't see that finger at all (as in the second image in your question; the left index finger is obscured), or we can see that his left index finger is noticeably stumpy (click to embiggen all images):

• At Aragorn's coronation

  

• At the Green Dragon

  

• At Sam and Rosie's wedding

  

• Writing in his diary

  

• At the Havens

  

That being said, I've heard it claimed that Elijah Wood's finger is visible later in the Mount Doom scene, which is possible (though I struggle to see it; I suspect most people would, unless they were specifically looking for it); consider (as always, click to embiggen):

However, even here it's clear that the intent is to show the finger having been removed; this, as with any later shots where the finger is visible, is just a production mistake.