I haven't finished my re-read of the Lord of the Rings trilogy (I just finished The Two Towers), but, at least in the movies, Samwise Gamgee consistently refers to Frodo Baggins as "Mr. Frodo."
Why is this?
I noticed that Merry and Pippin call Frodo and Sam by their first names only, and vice versa, so why is Frodo bestowed with the formality of being called "Mr. Frodo"?
There are some good answers here already, but some of the comments betray an unfamiliarity with the social system that Tolkien was portraying in the Shire.
The Shire is very specifically based on the rural English Midlands of the late 19th century that existed, although already starting to disappear, when Tolkien was growing up. In that society, there was a very clear distinction between the different classes. Frodo represents the landed gentry - not actually aristocracy, but powerful and influential as a result of owning large amounts of land. It's not made explicit, but as owner of Bag End he was almost certainly also the owner of much of the land about. The Gamgees may well have been his tenants.
Meanwhile, Sam is very much of the servant class. If you've seen the TV series Downton Abbey, for example, you can picture the sort of character he's meant to represent. Being "in service" was regarded at that time as a very good occupation for someone like Sam. He became very close to Frodo, and devoted to him, but the relationship was never one of equals. Sam would never have dreamed of referring to Frodo without the honorific.
Even Faramir, from a completely different society, recognizes this relationship, and tells Sam:
"Patience!" said Faramir, but without anger. "Do not speak before your master, whose wit is greater than yours."
The Lord of the Rings - Book IV: Chapter V - The Window of the West
After Frodo sails from Middle-Earth at the end of the book, he leaves Bag End to Sam. At that point, Sam is himself propelled into the gentry: he becomes Mayor, and founds a dynasty of his own.
Mr. Baggins already belongs to Mr. Bilbo Baggins, so it's only natural to come up with a different honorary reference for Frodo.
Recall that Sam is Frodo's gardener, and he was the son of Hamfast (the old gaffer), Bilbo's gardener before him. He grew up being the son of, and then himself, a servant of Bilbo's and Frodo's, and so likely called them "Mr. Bilbo" ("Mr. Baggins"? possible but I doubt it) and "Mr. Frodo" all his life.
One of the things that JRRT explains in his other books is that he tried to explore social interaction through the use of language. So the way that Sam addresses Frodo especially reflects the fact that Sam's role is subservient, and also his presence on the journey is under the sufferance of Elrond.
The phrasing is not intended to reflect modern-day servitude, as much as it is to acknowledge Sam knows his place in Hobbit society, that being, in this case, to look after Frodo. Sam's use of language ( which, remember, was translated from the Book of Westmarch ) reflects as much his accent as his style of talking. It is not unlike someone from the West Country; if you translate their language directly, it can sound subservient.
It is always worth remembering that Tolkien's motivating force was language, so wherever he uses language, he has good, clear, and usually historical reasons for doing so.
There is another relationship that Sam and Frodo represent - that of officer and enlisted man/batman that Tolkien would have been very familiar with. In England officers were usually upper-class and enlisted not, but the enlisted man had a wealth of experience that no good officer would overlook. And the batman would take care of the officer in a physical sense as well.
Think a young lieutenant talking with a veteran Sergeant. Sure, the Sgt. must salute the Lt, but the Lt would be a fool not to respect the Sgt. The officer tells the Sgt. what the goal is, and the Sgt. knows how to carry it out from there. Think of all the movies where John Wayne plays a Sgt. He's not just an underling. Both are playing important roles.
Frodo, Merry, and Pippin are all effectively landed gentry in the Shire, while Sam is just a servant / gardener. Such class distinctions were still quite prevalent in the early 20th century in Britain when the books were written.
First of all, Sam was Frodo's gardener. Frodo kept the Ring hidden for 17 years (not in the movie). Sam grew to know his employer, respect him, and form a friendship. He valued Frodo and their friendship a lot, and looked after Frodo, considering himself Frodo's friend and guardian.
Sam is also a lot younger than Frodo, who is about 50 years of age in the book. Sam was born in 1380 of The Shire reckoning, while Frodo was born in 1368. Frodo looks young in the movie, but the ring makes the bearer stop aging (maybe the movies took this into consideration). Frodo stopped aging for 17 years while he kept the Ring hidden, similar to Bilbo.
The portrait of Sam in LOTR is of a young servant, inheriting his father's craft. Bilbo teaches Sam how to read and write. The position of Sam to the positions of Frodo, Merry and Pippin is not equal in Hobbit society, this is not portrayed in the movie.
The book also mentions Frodo, Merry, and Pippin being old friends, while Sam is not mentioned in that way until much later.
Quotes from the book:
As the years passed, Frodo seemed to stop aging and appeared at age 50 like a robust Hobbit just out of his tweens. Frodo was red-cheeked and rather stout, but taller and fairer than most Hobbits, with brown hair, bright eyes, and a cleft in his chin.
Sam had learned to read and write from Bilbo Baggins and he listened eagerly to Bilbo's tales about his adventures, particularly the ones about Elves.
