In terms of the definition outlined in the first post, gamebreaking is:
#1. Rogues. You can argue about it all you want, but the fact of the matter is that rogues are the most powerful unit in the game, hands-down. It's somewhat ironic that they're completely useless on their own, but you're still leagues ahead of anyone without them when it comes to PvP combat. That free first hit is invaluable and that's almost an understatement in itself. Inno could change this situation in an instant by merely switching all combat AI to match the continent map, but that would obviously be an extremely controversial change even if removing the prominence of rogues from combat may be better for the health of the game.
#2. Being able to level GBs past 10 can be gamebreaking in certain cases. Chief among them being:
#2a. The Arc has permanently changed everything related to increasing GB levels. Occasionally people complain that the Arc is 'useless' for people in AF and OF but the treasury bonus could be completely removed from this GB and it'd still be the most powerful GB in the game. FP flows like a waterfall for everyone who's taken the time to level one up, and it snowballs into being able to rapidly level up any other GB of the owner's choosing, though heavily dependent on how much the owner utilizes this bonus by reaping rewards from the GBs of others. I believe the developers tried to have some foresight* and curb the impact the Arc would have on the game, but it wasn't even nearly enough. This could also easily be remedied by changing the Arc's bonus, but that wouldn't undo the impact that it's already had on the game and, like with rogues, would be a very controversial change.
*The Arc is the only GB with a bonus that has 'breakpoints' at fixed levels (levels 58 and 80 both lower the amount you get per level past that point). Every other GB either gains the same amount for every level past 10 or it very gradually increases/decreases the further you go.
#2b. The Château Frontenac is also gamebreaking. When sufficiently levelled, the coin and supply rewards from quests will be boosted to a point where they can feed into a cycle of quest looping that can continue nigh-indefinitely, providing dozens of goods and medal rewards which are also boosted by the Château's bonus. Eventually one can reach a point where the cycle is limited primarily by the owner's willingness to keep clicking through the quests, rather than the stockpile of coin and supply they have. This one could be fixed by having only recurring quests be unaffected by the quest reward bonus.
#3. Diamonds are technically gamebreaking as well, but as these are what keep the lights on in the server room there isn't a whole lot to do about this one. More irony here in that the above examples are even more gamebreaking than the 'pay to win' currency in the game.
Summary of ways to fix gameplay: Remove rogues and cap GBs at level 10 once again. Simple!