In an ideal world, people would develop the ability to adapt to a change or decisively figure out a way to counter something deemed as too powerful. In League of Legends' case, the unfortunately truth is sometimes things are just too powerful. Let's take a dive into some candidates for this category.
The New Black Cleaver
Alright, let's be honest. This meta-breaker hasn't had the longevity of some other things that will appear on the list, but the impact it had on the game was too great to have it excluded. Though a bit more tame in its current form, the entire LoL community spent a week of agony coping with this problem. Wasting no time to abuse it to its fullest, AD Assassins would have every stat they possibly needed with 2 of these monstrosities, be tanky enough to survive an engage, and still remain fully capable of making the life of an enemy carry living hell.
One would argue that this is the role of an assassin, but if you're going to be designed as a glass cannon, the "glass' adjective should remain and the CDR makes them more than slippery enough if the 500 HP worth of durability doesn't cushion their mistakes enough.
Rengar
When developing our candidates, we tried to avoid simply listing a champion as a viable choice. But seriously, just...Rengar. Curiously, most of his problems root from not being utilized in a way Riot would allegedly want him to be used. Intended as an assassin, Rengar was first uitilized as a burst mage, and then transitioned into an HP stacking tanky machine that would utterly devastate anyone if he were to pull ahead of the opposition. His percentage heal was an enormous problem which further enabled the abuse of flat HP builds.
If that wasn't enough, Rengar was also top-tier at split pushing due to his waveclear, his dueling capabilities, and his ability to stealth and walk away from any careless situation. Rengar was a huge problem in League of Legends, and it's astonishing that he was ever left available in a competitive environment.
The Holy ADC Trinity
Not so much "broken" as "horribly boring," everyone who stays tuned to LoL E-Sports regularly is very familiar with Ezreal, Corki, and Graves. To an extent, the same is true today, though with much less severity than it once was. There's perfect reason that these characters were powerful. They all had poke, a strong laning phase, a repositioning tool, great base stats, excellent steroids, and excelled in late game simply because of the multiplicative strength of AD Carry itemization, even if not as much as their hyper-carry counterparts.
Though Graves was much less commonly seen, he was still a part of the problem. Primarily, however, lanes in tournaments quickly were absurdly vapid Corki vs. Ezreal fests. For a lot of bottom lane enthusiasts, this was driving people mad, and thankfully is slightly less of a dilemma than it once was.
Pre-Nerf GP10
To open this topic, I would like to clarify that this is not about everyone buying Heart of Gold. There was a time this year where it was much, much worse, and unfortunately, very viable for 4 out of the 5 characters to kick-start their build with Heart of Gold, Philosopher's Stone, and Kage's Lucky Pick in virtually any scenario. It did weaken the laning phase, but the sacrifice was not significant enough to put a heavy damper as to make the trade-off not worthwhile for the dividends provided late game.
There was counter-play, albeit it being extremely unforgiving. You could all-in your opponent with a very early-game aggression oriented build such as Doran's stacking. Unfortunately, if this didn't pay off, you would fall even further behind. Gold Per 10 was "toxic" (if I may use a hype phrase) in the sense that it was too potent and begrudgingly forced the opponent into unfavorable situations due to the strength of the items and the utility they brought.
Global Ultimates
Nothing was quite as irritating as the constant fear of pressure despite the enemies' location. Though slightly remedied by the changes that would render the globals as only "partially global", it still was overly dominant. Highly contested picks often involved those with these abilities, such as Shen, Karthus, Twisted Fate, Soraka (early on), Ezreal, and Nocturne. Many of the picks and bans were specified around the characters listed prior; You picked Nocturne to counter gank Shen or Twisted Fate. You (used to) pick Soraka to counter Karthus. Ezreal, in general, was too strong on top of that global ultimate. Ideally, you don't want to fight fire with fire because fire is the most potent; that is not balance. The fact that the global cooldowns were much shorter earlier in the year was not helping the matter at all, either.
Though still highly contested, it's not as much of a problem as it is a tactic now. Still, the success ratio of teams winning the lategame due to who has more globals has drastically reduced compared to how the problem once was.
Oracles-First Junglers
Professional players might agree on this being one of the most game-breaking discoveries in the old metagame. The tier of junglers was heavily influenced by their ability to operate on a budget efficiently, and thus, the junglers capable of carrying Oracles early and safely. If a jungler got too far ahead early, you can take a healthy gander that a team properly utilizing it would allow for no comeback and they would be infinitely trapped behind. Consider the fact that the first roamer with an oracles would immediately put the other team in a position where even the simplest of tasks, such as farming at the tower, were not a safe approach.
If you killed a jungler with oracles, your team would be at an unprecedented disadvantage due to both the immense strength of the item as well as the enormous monetary setback of the already exceedingly low income the junglers acquire. It was a very messy situation when professional games would often come down to who could get away successfully with mobility boots and oracle, developing a ruthless monopoly on map vision.
