Assassin's Creed II was much more strongly received than the original game as an almost-entirely across-the-board improvement on the original in game terms, plus introducing a beloved protagonist in Ezio Auditore da Firenze.
Borderlands, despite being a mechanically sound fusion of FPS and RPG elements, was rather bland and repetitive until the DLCs.
One of the major complaints about the base game was that it took itself too seriously, unlike the expansions, which had far more humor. Also, judging by what has been seen so far of the sequel, Gearbox has decided to take the humor and general nonsense Up to Eleven.
The Burnout series started out poor, with the initial game being not very well received. Its second installment was better, though it was really the third title that could be considered the beard-growing moment, with the games addition of the Takedown maneuver.
City of Heroes. Initially a passable MMO whose main strength was the amazing costume editor, the addition of a mission creation tool to allow players to create their own instances for other players to enjoy took the game to the next level.
The mission creator is recent, but many players will also attest that the game has steadily and massively improved since launch due to major rebalancing that actually worked (despite copping some rage at the time), addition of many costume parts, improved writing, and a general dev focus on player-friendliness. While it's hard to place a particular turning point, the release of City of Villains could be considered a major game changer.
There are a number of changes which qualify, but the most marked turning point is probably the sale of the franchise from Cryptic to NCSoft, the creation of Paragon Studios, and the departure of much-loathed original lead developer Statesman. Several changes prior to this change were pretty good (particularly the City of Villains expansion and the addition of an an economy in Issue 9), but most of the real solid improvements came afterward (including weapon customization and the ability to play arcs you've outleveled through time travel in I11, I14's aforementioned Architect system, power customization in I16, and the recent Going Rogue expansion).
The Dark Forces Saga started off as a standard FPS clone of Doom with a Star Wars theme, but its sequel Jedi Knight introduced force powers and lightsabers. It eclipsed its predecessor and set the flavor of the series from there, to the point that "Dark Forces" isn't even used in the title anymore (and creating one of best known cases of Colon Cancer ever).
In 2008, Korean game developer Pentavision took their DJMAX series of Rhythm Game s to the arcade environment under the name DJMAX Technika. Drastically different in gameplay from its predecessors, the game was already well received, although hints of Fake Difficulty, Obvious Beta and general needless complications were present. Come 2010, the sequel, DJMAX Technika 2 improves on the previous game by being more streamlined in every way, adding new songs (half of which are revivals of classic songs in the franchise) and new modes such as the every popular Crew Race.
The first two Duke Nukem games were fairly unremarkable 2D side-scrollers. With Duke Nukem 3D the genre changed to first person shooter complete with innovate weapons, impressive (for the time) level design, and a raunchy sense of humor. As a result Duke Nukem has become one of the most famous video game heroes of all time.
Duke Nukem 3Ditself grew a beard about halfway through its development, as it was originally intended as just a 3D continuation of the original platformers with none of the humor and personality that the final version would be famous for. This version, retroactively dubbed LameDuke, was mercifully put to rest and eventually retooled into the game we know and love today.
Dungeons and Dragons Online This game grew a beard in September 2009 when the game went to a Free to Play model and Turbine sued Atari (the digital rights manager of D&D). All the updates since Update 9 have featured much better quest designs.
As of Skyrim though, the series seems to have grown back the beard a bit. it has been more universally better received on release by fans and critics.
Whilst there may never be a consensus as to what point this happened in the series, Final Fantasy is usually credited with this. Some commonly invoked examples are: FFIV, when a Final Fantasy with set characters first arrived in the US, as well as being the first on the SNES; FFV, which was the first time the series looked at itself from a retrospective point; FFVI which is often seen as the culmination of all the lessons learned in FF's I-V; FFVII which had the greatest duality of Sci-Fi vs. Fantasy in the series; or sometimes even FFT, which was among the first FF's to use very complex and conflicted characters who developed over the course of the game. Remember, of course, that the fanbase will never agree over this.
'Final Fantasy XIII-2 seems to have grown a beard on its predecessor, which takes some doing, since it's basically the Revolutionary Girl Utena of the Final Fantasy franchise. A world with real depth, dramatic plot pacing, and npcs you actually give a damn about had to wait until the sequel, apparently. Also: "You didn't like XIII's linear gameplay? Here, have a quadruple tesseract playing twister with a Klein Bottle...mwhahaha!"
Fire Emblem grew the beard with Genealogy of the Holy War, the fourth game. It featured a much darker story than the past games, along with larger maps, more character interaction (to the point of being able to pair people up), and tons more drama.
While the gameplay of Gears of War had always been good from the start, the plot of the campaign didn't really start to develop beyond a blatant Excuse Plot until Gears of War 2.
The first two Grand Theft Auto games were reasonably popular due to their unique gameplay and controversy, but once the change to 3D went full in Grand Theft Auto III, it was an immense hit, and essentially one of the first well-done open-world sandbox games. Of course, most players were already thinking "This would be even better in 3D."
The first Guild Wars campaign, Prophecies, is generally considered to have long, monotonous levels compared to the other games. It also had very little max-level content and was almost completely serious, when reaching max level early on in the game and having a slightly dry sense of humor would later become part of the games signature style.
Factions, the second, solidified many ideas that would be re-used in the next installments, including henchmen with distinct stories who talked to the player, a "starter island" that the character generally leaves around max level, and several pacing adjustments. It doesn't hurt that the Factions era is still widely regarded as the most balanced PvP environment in Guild Wars history.
Heroes Of Might And Magic V started out as a 3D-remake of the third game with awkward translation and mostly lazy cutscenes (using the existing animations of heroes and units instead of moving the mouth). It grew its beard over the course of the two expansions, particularly the second.
Mario Kart revolutionized mascot racing during its time by having 8 characters in a race and having tracks with complex designs and obstacles. Mario Kart 64 took it a step further when the series made the leap to 3D by having better designed tracks, more items, and up to 4 people could play the game on the same console. Mario Kart DS led the way for online play for the series.
Though critics loved it unconditionally, the gameplay of the Mass Effect franchise was pretty rocky. 1 was a straightforward third-person run-&-gun somewhere between Halo and Doom, hampered by unnecessary Take Cover elements. 2 put cover-based shooting center stage, but adjusted all the game's other elements (gun damage, special abilities, HP totals) a bit too far, resulting in fragile and unbalanced gameplay. The trilogy's final entry managed to mix everything just right, to the point that 3's multiplayer mode, at first derided as a cash-in mechanic, is now enjoyed unironically by franchise fans.
Mega Man Battle Network was received rather warmly, but is an Obvious Beta to even the most biased of critics. Its sequel introduces the Style system, a wealth of replay value in the sidequests and expanded chip library, the much appreciated ability to flee from battle without the need for a specific Battle Chip and a more varied soundtrack, among other things, creating an overall more polished experience.
Neverwinter Nights started out with a rather boring story, full of Plot Holes, with about 2 three-dimensional characters in the entire game. It started growing its beard with the expansions, but the real potential of the engine didn't really emerge until the greatest works of toolset manipulation (A Dance with Rogues, Sanctum of the Archmage, The Bastard Of Kosigan, the Shadowlords series, and more) started to show up.
Persona 3 was this for the Persona series. While the others were certainly good games, this was the installment that made the series famous. It was the first to come to the US without having been butchered by borderline racist translators and also introduced the social link system.
Not just Persona. This is considered to be the game that made the entire Shin Megami Tensei series big in the west, and Persona 4 proceeded to make it even more well known.
Phoenix Wright: Ace Attorney: The first two cases of the first game are relatively simple, designed with fairly over-the-top characters while the player learns basics of gameplay. In the third case, the HSQ shoots up, along with Character Development, and, suddenly, the game switches from "non-stop wackiness" to one with real stakes. The next game in the series doubles down on both the Rule of Funny and Earn Your Happy Ending elements that made the first game so successful.
Pikmin 1 and 2 were considered cute classics for GameCube. Pikmin 3, meanwhile, was highly publicized by Nintendo and regarded as one of the Wii U's best games.
Resident Evil 4 was far, far less of a joke than the previous few games of the series due to the massive changes in controls, camera, setting, everything, and wound up winning Game of the Year at quite a few websites and magazines. The gameplay and puzzles were more acclaimed than in the PlayStation titles (which themselves weren't too bad)
Even before that was Resident Evil 2, which, rather than having a B-Movie story like the first game, ended up having one of the most engaging plots in the series. Later, the first game was remade in 2002, doing a few Retcons and trying to connect it to the later installments which made it seem a lot less cheesy.
Sam & Max: Freelance Police: Season Two is widely considered to be much, much better than Season One, due to having harder puzzles, more variation in locations, more unique episodes, a much more coherent story arc involving most of the supporting cast and killing off the Soda Poppers.
Season One also grew the beard with Abraham Lincoln Must Die. It's no coincidence that this is the episode that's given out for free (for good reasons).
The Devil's Playhouse (Season Three) is also this. A darker, much tighter storyline spoofing the works of H.P. Lovecraft and The Twilight Zone, with top-notch writing, acting and humor. Max gaining some rather fun psychic powers and The Soda Poppers staying dead certainly helped too.
The Sega Superstars games only started to pick up praise from the critics with the third installment, Sonic & Sega All-Stars Racing.
There is widespread agreement among Silent Hill fans that the beard for the series was grown during Silent Hill 2. Given that the first game tends to be listed as a favorite even after eight games, this is a testament to how revered the second game is within the fandom.
The series has long suffered from Broken Base Syndrome, with later games being generally less well-regarded. There are some dedicated purists who refuse to ever acknowledge any game after the fourth, but among the more open-minded fans, Silent Hill: Shattered Memories has achieved a cult following and some think of it as when the series grew its second beard. That latest game has been quite better received critically and among fans than several games previous speaks of some truth to the idea.
Sonic the Hedgehog was about a blue, 90's cool anthropomorphic hedgehog fighting against a mad scientist who wanted to use woodland creatures to power his robot death armies. The Chaos Emeralds were already present, but Sonic only collected them in order to stop Robotnik from taking them. The "Best Ending" occurred if you collected all seven emeralds by diving into a dizzying, multi-colored bonus area. Enter Sonic the Hedgehog 2. Sonic now has a mutant fox best friend who can run just as fast as he can, taught Sonic the Spin Dash, which today is just as much a signature move to Sonic as the Ground Pound is to Mario. Sonic and Tails are both playable, allowing for 2-player co-op, and can even race each other in a bonus stage to get the most rings, avoid bombs, and collect the emeralds. Oh, and collecting all 7 emeralds turns Sonic into Super Sonic.
Sly Cooper and the Thievius Raccoonus was an episodic, linear game where you played through level after level until you faced a boss and moved on to the next episode. In the sequel you instead perform "jobs" which can be anything from sabotaging the weapons of the Big Bad to recon to setting up an escape route, all building up to one final heist where you stick it to the bad guy. Plans can go awry and force you to improvise, there is plenty of World Building and exploration, and much more emphasis is put on the Gray and Gray Morality of a storyline where the protagonists and antagonists are both criminals.
Star Control was a relatively popular turn-based strategy game including a spaceship melee mode a la Spacewar!. The sequel, Star Control 2 increased the scope of the original with a story-driven adventure mode and various other elements. Star Control 2 went on to become widely considered one of the best video games of all time. Alas, there was noStar Control 3.
Super Robot Wars: It started off on the Gameboy with just three series showing up, no pilots or plot at all. It jumped to the Super Famicom and the 2nd had an excuse plot, with the 3rd starting up gags from the various series and doing retcons for disliked plotlines. This continued on for a while as a fun excuse plot series but people say that with Alpha starting the massive and complex story is where the franchise started perfecting.
Tekken was pretty much a direct arcade port with a versus mode, some remixed music, and Galaga. It was a decent fighter, and an impressive one at that, but it left a little to be desired. Then came it's sequel. New characters, tons of new modes, better emphasis on story, better music, better graphics, the works! And the series would only continue to improve with the third game in the series.
For most Wario Land fans, Wario Land II (the second Game Boy title, not the Virtual Boy one) did this, as the series gameplay diverged significantly from that of the Mario series and actually started developing its own identity with things like the transformations. Which of the next two games is better on the other hand, seems to be dependent on the player.
In a hardware example, the PlayStation 3 started out as a laughing stock with a ridiculous price point, very few quality exclusive titles for well over a year after launch, and a fair amount of meme-generating idiocy (Real Time Weapon Change, Giant Enemy Crab, Attack Its Weak Point and For Massive Damage) by Sony's PR department. It was also notoriously difficult to make games for, keeping away third-party support. Things began improving in 2008 with the release of Metal Gear Solid 4: Guns of the Patriots, which demonstrated what the system was capable of. By holiday 2009, they had launch a new slimmer model, dropped the price in half, and started a new campaign of genuinely funny advertisements featuring Sony's fake Vice President of Whatever-The-Hell-He-Wants-To-Be-VP-Of Kevin Butler. The fact that the PS3 had finally developed a very respectable game lineup didn't hurt either.
The PlayStation 4 is this for the brand overall. After the past two generations with increasingly frustrating-to-code hardware and inflating ego that was finally crashed down like the Hindenburg , Sony finally decides "hey, let's ask what the developers want". Not only did they win the developers over again, but Microsoft's less-than-stellar reveal of the Xbox One was heavily taken advantage of by Sony to win the gamers at E3. And it shows. The PlayStation 4 is the fastest selling console in history.