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I believe that in the 2nd movie, Merry and Pip meet Treebeard and drink ent-draught from Fangorn. They didn't appear any taller when we see them in ROTK though, very strangely they are the same heights as Sam and Frodo, and Pippin is shorter than Sam!

I know that they appeared taller (and noticeably stronger in Battle of Bywater) in the books, but how come they are not in the movies: was the movie's ent-draught lagging? How come Merry and Pippin appear the same height as Frodo and Sam in the movie?

Based purely on conjecture, I believe it's because the scene where the ent draught is taken, only appears in the extended release.

While it still happens, for the cinema release it wasn't there, so, it would have been rather strange for the audience to have Merry and Pippin be suddenly and inexplicably taller.

Of course, perhaps they should have changed it for the extended release, but evidently they didn't.

TL;DR: Because of the way the film versions were cut, making them taller would introduce a continuity issue to the theatrical release of the film.

In the Films

The scene you describe - where Merry and Pippin are drinking the Ent-Draught - was only included in the Extended Edition of "The Two Towers". Like the original scene in the book, the dialogue in the film scene (referred to as 'Drinking the Ent-Water') does indeed make reference to the hobbits growing taller by a few inches:

Merry wakes up and looks around. He sees Pippin sitting by a barrel next to a waterfall drinking.

MERRY: Hello? Treebeard? Where has he gone?

PIPPIN: I had the loveliest dream last night. There was this large barrel, full of pipe-weed. And we smoked all of it. And then... you
were sick. I'd give anything for a whiff of Old Toby. stretches

Merry hears what sounds like trees cracking.

MERRY: Did you hear that? There it is again. Something's not right here. Not right at all. Pippin stands up, and when he does he makes a
strange noise.

MERRY: (looks surprised) You just said something...Treeish.

PIPPIN: No, I didn't. I was just stretching. (makes the noise again)

MERRY: (walks round Pippin inspecting him) You're taller.

PIPPIN: Who?

MERRY: You!

PIPPIN: Than what?

MERRY: Than me!

PIPPIN: I've always been taller than you.

MERRY: Pippin, everyone knows I'm the tall one. You're the short one.

PIPPIN: Please, Merry. You're what? Three-foot-six? At the most? Whereas me, I'm pushing 3'7". (he burps a tree sound again) 3'8"!
(he takes another drink quickly)

MERRY: Three-foot-eight. You did something.

When we see the four hobbits side by side later, these new inches aren't there (as seen in your image). That's specifically because the Ent-Water scene had to be cut to meet time restraints, and having Merry & Pippin taller at the end with no explanation for it would confuse audiences.

In 2002, Peter Jackson stated in an interview with EW that he still wanted to include the scene in the Extended Edition because it's funny:

Q: What was the most painful thing to cut from this one?

A: Not a great deal was cut. There’s not very much [extra footage] with Frodo and Sam because there’s so much of Gollum in each of their
scenes, and because those shots are so difficult to do, you don’t
really do Gollum unless you absolutely know that it’s going to be in
the movie. [We cut] the whole scene where Merry and Pippin drink the
Ent draught. I’ll definitely put that on the DVD because it’s so
funny.

Now, all of that said, you could argue that Pippin, at least, DOES seem a couple of inches taller in the image you posted (vs his earlier appearances). The height of all four Hobbits isn't consistent throughout the trilogy - sometimes they all seem the same height, as in the image you posted, while at other times they seem to have varying heights. Frodo is generally shown to be the tallest of the four in such scenes, with Pippin being the shortest.

Here's an example from the cornfield scene in "Fellowship of the Ring":

And here's another from the Rivendell scenes of "Return of the King":

Note that in the latter scene, which occurs after the eagle rescue but before the ceremony in Gondor which you posted, Merry & Pippin now both seem taller than Frodo & Sam - Merry by a few inches. This inconsistency may indicate that the height change was originally included with the Ent-Water scene, or it may simply be odd angles and/or something missed by the editors.

In the Books

In the books, the height difference was much more drastic. Merry and Pippin ended up being the tallest Hobbits in the history of the Shire, with Merry still the taller by a small amount (perhaps an inch or less) and approaching almost five feet in height.

Prior to their consumption of Ent-Draught, the tallest Hobbit on record was Bandobras Took at 4'5". So even had the film included the height difference, the paltry few inches mentioned by Pippin was much less than the almost two feet taller they grew in the book.

Or, as it says in the LotR Prologue (Page 2 of 'Concerning Hobbits'):

For they are a little people, smaller than Dwarves: less stout and
stocky, that is, even when they are not actually much shorter. Their
height is variable, ranging between two and four feet of our measure.
They seldom now reach three feet; but they have dwindled, they say,
and in ancient days were taller. According to the Red Book,
Bandobras Took (Bullroarer), son of Isengrim the Second, was four foot
five and able to ride a horse. He was surpassed in all Hobbit records
only by two famous characters of old; but that curious matter is dealt
with in this book.

I believe in the extended version they do drink ent-draught and I do think they become taller due to the dialogue shared between the two:

MERRY: (walks round Pippin inspecting him) You're taller.

PIPPIN: Who?

MERRY: You!

PIPPIN: Than what?

MERRY: Than me!

PIPPIN: I've always been taller than you.

MERRY: Pippin, everyone knows I'm the tall one. You're the short one.

PIPPIN: Please, Merry. You're what? Three-foot-six? At the most? Whereas me, I'm pushing 3'7". (he burps a tree sound again) 3'8"! (he takes another drink quickly)

MERRY: Three-foot-eight. You did something.

However, near the end of the The Lord of the Rings: The Two Towers the two have went with Treebeard to Isengard and have succeeded from acquiring it back from the Orcs, both Merry and Pippin can be seen with half of their bodies concealed by water. Merry checks his height against Pippin's and is quite happy that 'the world is back to normal' upon finding out Merry is back to being taller than Pippin. To me, this meant that the effects of the ent-draught was only temporary:

MERRY: (Checks height against Pippin's)

PIPPIN: (Turns around to face Merry) What are you doing?

MERRY: Nothing, worlds back to normal. That's all.