Animorphs at first did it with the third book in the series; while the first two had helped to establish the core plot and the setting, the third book took a more unique turn, centering around Tobias, the most mysterious member of the group who in the previous books had been trapped in the form of a hawk. Other points later in the series' 54 book run could also be considered growing the beard, depending who you ask. Perhaps when Marco's mother is revealed to be Visser One, when the conflict escalates to a full scale war in the later books, and more gradual as the characters grow more mature over time. There is also a very notable beard-growing for the companion books such as the Andalite Chronicles and Hork Bajir chronicles, with much more mature and engaging storylines following on characters on exotic alien worlds. On the other hand, some fans argue that the later books in the core series saw a decline in quality, where Applegate had many of the books ghostwritten (though she heavily edited them to fit), and in the climax of the series where some were upset at Rachel and Ax's deaths.
It's telling that the only book in the mid-to-late range that is generally well-received by fans is a take on "The Enemy Within" which is also the only book not ghostwritten until the final two.
By book five in the Lemony Snicketseries, the formula has started to feel a bit tired. While the characters remain strong and the events remain unfortunate, the end of The Austere Academy is the first time we hear the letters "V.F.D.". From there it escalates from a deconstruction of children's adventure novels to "My Very First Dostoevsky".
The Aubrey-Maturin series picks up considerably with 3rd and 4th books HMS Surprise and The Mauritius Command, after being given command of the titular Cool Ship and heavily reducing the land-based romantic storylines of the 2nd book Post Captain.
Lord Foul's Bane, the first book in Stephen R. Donaldson's Chronicles Of Thomas Covenant series, isn't very good. The series improves dramatically in the second book, titled The Illearth War, and stays that way.
The Clique started growing the beard with These Boots Are Made for Stalking and My Little Phony. The preceding books had varying degrees of quality at best, stagnant characters suffering even further from Flanderization and flat-lining plots. In particular, the series gained a reputation for having an Audience-Alienating Premise; it was series about an Alpha Bitch and her Girl Posse, and further mired in a reputation of Darkness-Induced Audience Apathy for its unlikable, backstabbing or otherwise despicable characters. Starting with these two books, characters started experience actual, meaningful character growth and actual maturity, the main protagonist (or what passes for such) started standing up for herself and in short, the series actually became what people were expecting and hoping for. Unfortunately, it was too little too late as These Boots Are Made for Stalking and My Little Phony were the third and second-to-last in a series spanning nearly two dozen, and The Clique is now considered by many to be Deader Than Disco.
The early Discworld books (at the latest, up to Small Gods) felt far different than their latter counterparts. Particularly glaring within the separate section of the Disc mythos: compare and contrast the Granny Weatherwax from Equal Rites to the one in Carpe Jugulum. Or the Lord Vetinari in The Colour of Magic (Word of God had to step in and confirm that it was the same Patrician, and not one of his thoroughly insane predecessors) to the Magnificent Bastard of the Moist Von Lipwig books.
Likewise, the first two Discworld books are straight parodies of Sword And Sorcery fantasy. The series began to grow its beard in Equal Rites and Mort, where it went from a parody of fantasy settings to using its fantasy setting to parody everything else.
The Dragonbone Chair, the first book in Tad Williams' Doorstopper fantasy series, Memory, Sorrow, and Thorn, tends to drag on and doesn't introduce the main antagonist until several hundred pages in. Once the series gets going, it's very good, but you still have to get through much of the first book to get to the good stuff.
This seems to be the case with most of Tad Williams' doorstoppers. The protagonists only know that their lives are going to hell; they don't know why, there are webs within webs, etc. Awesome characters, storytelling, worldbuilding, and prose keep this from becoming the problem it would be in the hands of a less capable author. But it's a given that you will have no idea what's actually going on until the last five hundred pages or so.
Incidentally, the main hero grows a literal beard in the course of the story, symbolizing his significant maturation from a lazy kid into a worthy king. The cover art makes the difference especially striking.
According to Jim Butcher the first book was written to show a creative writing teacher how bad her method of writing was.
While Alien Bodies, Seeing I and The Scarlet Empress are all individually good books, the Eighth Doctor Adventures doesn't really hit high gear until the introduction of fan favorite companion Fitz, and (arguably) the departure of sometimes-Scrappy Sam. With the addition of a second companion to shake up the Doctor-Sam dynamic, as well as the introduction of several Myth Arcs, the books became the truly phenomenal, often times groundbreaking spin-off material they're known for now.
Interestingly enough, while the introduction of Fitz is generally considered to be behind the EDA's beard growing, his first book, The Taint, is considered fairly weak and new fans are encouraged to skim it for Fitz's scenes. If anything, it's Unnatural History, which turned Sam's character arc on its head and moved the Faction Paradox arc to the foreground, that can be considered the definitive moment that EDA became awesome.
To some, Galaxy of Fear was much improved after Army of Terror, when the Big Bad of the first six books was defeated and the formula changed - also, in the subsequent books DV-9 had been Put on a Bus and the relationship between Hoole and the kids was much less strained.
Gardens of the Moon, the first book in Steven Erikson's gargantuan Malazan Book of the Fallen sequence, drops the reader in the middle of an ongoing war with little explanation of what is going on. The lack of scene-setting or explanations for concepts in the book have led many to give up on the novel, as acknowledged in later editions by the author. Fans suggest that the book doesn't settle down and become comprehensible until a good 150 pages in, and many suggest skipping it and starting with the more traditionally-structured second book, Deadhouse Gates (set on a different continent with different characters) instead.
The Prisoner of Azkaban, the third book of the series. The first two were fun, wonderfully-painted page turners with magnetic characters but didn't seem to be much more than that. Then book three's title character, who was mentioned very briefly at the start of book one, is brought into focus and it just builds from there, making it clear that this isn't just some fluffy kid/teen series but an incredibly intricately plotted seven part Myth Arc.
An even more noticeable instance of the trope is the fourth book, Goblet of Fire. Not only does it double the page count of the previous books, in fact tripling it over the first book, the climax is the turning point of the series. Up until then, it had been lighthearted and innocent for the most parts, but after this book, the series takes a swift turn into dark and grim territory, with a high body count, and adult themes and concepts introduced.
On the author's part. While it may not be extraordinary, The Host was much better received, and overall is much better than Stephanie Meyer's other works, and is perhaps an indication that she can indeed write things other than novels about sparkly vampires.
Ian Rankin, acclaimed Scottish author of the Inspector Rebus novels, started out the series with quite straight forward serial killer and murderer hunts. The fourth novel, Strip Jack, had a change in tone in dealing instead with the sordid life of a (fictional) British politician. Afterwards, the series began to focus more on the morally gray world of big business and British politics, and the relationships between the two. The series was much better for it.
A moment like this for The Lord of the Rings appears in The History Of The Lord of The Rings. We see several early drafts of the beginning of the book that would become The Fellowship of The Ring. They are all very similar, in construction and tone, to The Hobbit. The original villain of the piece was always intended to be the Necromancer mentioned in The Hobbit. However Tolkien's original interpretation made him a fairly light-hearted, almost mischievous villain. However, somewhere along the line (in a moment which, sadly, Christopher Tolkien could find no record of) Tolkien changed it so that the Lord of The Rings was actually Sauron, the villain from his Númenor legend. Tolkien then saw that the legend of the rings could be used as a way to explain how Sauron survived the destruction of Númenor. With the Smaug-like villain replaced by a character who was, for all intents and purposes, Satan, the book became more grounded and the story got much, much darker; growing into the legendary saga we have today. This dragged many other characters along with it - Aragorn, the wandering king-in-exile, was originally Trotter, a rather silly but experienced hobbit who dressed himself as a man and stood on stilts.
A surprising leap in style occurred between books six and seven of Ranger's Apprentice, with more originality, humour, and maturity in the following stories.
The first Rod Albright Alien Adventures book, Aliens Ate my Homework is a fun story about a boy dealing with a spaceship full of two-inch high aliens, hiding them from his parents and teachers by pretending they're toys and so on. The second book, I Left my Sneakers in Dimension X shifts the action from a generic small town to another dimension, revealing just how inventive Bruce Coville can be with aliens and settings, and that the plot of the first book wasn't so neatly wrapped up as it seemed. The final chapter reveals that Rod's father is apparently an alien and a former member of the Galactic Patrol. This leads to major Fridge Brilliance with regards to the first book - the aliens crashing through Rod's window was no accident, and their making him a deputy them may well have been down to his father having been 'one of the best'.
The first two Sherlock Holmes novels were huge hits in their time but many Holmesians agree that the Holmes and Watson that everybody remembers fondly didn't appear until "A Scandal in Bohemia", the first in a series of 56 short stories.
The Eyre Affair, the first book of the Thursday Next series, isn't bad, per se, but features disappointingly little use of the series' central gimmick of the title character being able to enter works of fiction and a comparatively conventional "stopping the bad guy" plot. Starting with book two, Lost in a Good Book, Jasper Fforde really went to town with the concept with nods to all kinds of different literature. Also, while some important plot threads are introduced in book one, the second really begins the series' fascinating juggling act between all its different subplots that frequently collide or call back to events several books previous in unexpected ways.
The Uncle John's Bathroom Reader trivia book series, beginning with volume 8 (Uncle John's Ultimate Bathroom Reader). This volume was was bigger than the last two volumes combined, and it started the gradual shift towards a more in-depth writing style. The evolution has continued in subsequent volumes, which now feature multi-part stories and an "extended sitting" section with even longer material. While the first few books had about 200 pages, the last few volumes have pushed 600.
David Brin's novel Sundiver, the first set in his Uplift Saga universe, is ok, but it is often recommended that readers skip to the second, the Hugo and Nebula winning Startide Rising, instead. This is made easier by the second book being set 300 years after the first, featuring a totally different cast and having minimal references to the first book.
The first book, The Eye of the World, cops a lot of flak for its intentional "borrowings" from Lord of the Rings and standard fantasy conventions. The second book, The Great Hunt, takes the story in a completely different direction and is much better, and the beard is completely grown in the third book, The Dragon Reborn.
After Jordan's death in 2007, young author Brandon Sanderson took up the reins using Jordan's notes, and his The Gathering Storm has another slightly different direction and is awesome in a refreshing new way.