There's no way Riot's going to introduce any method for permabanned players to try and recover their account. They've already tried something similar in the past, and it had an abysmally low success rate. 95% of the players Riot hand-picked and deemed "most likely to succeed" failed.
Beyond that, the permanent ban is supposed to be a punishment, and when someone gets a permaban, that's a sign that they genuinely didn't care about the rules, and Riot had to put their foot down. The point of the permaban is for Riot to get the message across that they will not tolerate people breaking the rules, and the consequence for doing so is appropriately harsh.
If Riot just allowed people to recover their accounts after a permanent ban, then the punishment would lose all of its impact and have no reason to exist in the first place. Why worry about losing your account when you can just play "nice guy" for a while and get it back?
The permanent ban is the absolute end of the road for an account. And, permanent means permanent. There's no way to get it back, and I don't see Riot implementing one in the near or far future, simply because of how easily such an option would be abused and how quickly toxic players would start ruining more games.
Besides, how many chances do you really need to understand that you should be following the rules? Riot feels like three minimum (or one at absolute minimum) works best. How many chances do you need? Four? Five? IMO, one chance is enough.
