Have you ever had a character live long enough after killing the big bad to become the villain?

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Last one. Doesn't seem to be a very popular topic.

Sarevok was always evil

This is almost a prequisite for experienced role-playing groups. I've had several former PC's become the BBEG, lieutenants to the BBEG, anti-heroes, or opposed to the party for their own reasons. It's a powerful narrative tool, but when used improperly can lead to severe, sometimes justified, That Guy behavior and strife between players and the GM. I've always found it's best to have the party's obligatory ambigiously moral CN character go bad, but I've had a few noteworthy successes with LG misguided Heroic Villains and power-hunger magic users, regardless of their previous alignment. Of course, it's something you can't use repeatedly, or twice in a row without care, otherwise the players will come to expect it. Best to keep it as an "ace up your sleeve," not something to rely on, but something to toss in when things are getting slow and you're three sessions away from pointless crawling that's boring to narrate and boring to play through. Like drugs, a little every now and then is fine, but too much, much too often ruins lives.

Tell us a story, user. What was the time that hit your players the hardest or surprised them the most? what methods did you use?
A good hero who just lived a life of suffering and became disgusted with the world?
A neutral hero in search of power who saw "good" was an impediment in his way?
A hero who beat the big bad and got to old age, eventually becoming hated and blamed for all the evils of society?

Through the course of a campaign I played in a few months back, my Mage lost his shit. Gained more and more power and it went to his head, slipping from CG, to CN, and finally down to CE. It was only when he got chewed out by a party member when he sobered back up and realized he was becoming a monster. He crawled back up to CN, and eventually ended the campaign at True Neutral. It was a hell of a ride.

Have I personally had a character become the villain? No.
Nor has it actually happened to any of my players, come to think of it.
So in short, no, I don't think it's not the only fate for heroes that don't die. Most of the heroes who live past their adventures tend to retire in my experience. I actually wrote a post on my blog about that recently, which i'll go ahead and link.
>draconick.com/2017/12/27/the-legacy-campaign/
That said, you could very easily make this turn to evil a common thing for your characters. It's a pretty standard trope, which is why you hear so much about it. Incidentally, I also made a post about that as well, so i'll go ahead and link that too.
>draconick.com/2018/01/16/character-development-fall-from-grace/

Well, my character turned into a friendly lich near the end. A player's dragonborn focused on fire magic became a partial antagonist during the next campaign we had though.

after we all broke out of prison for "political crimes" in our revival campaign, my guy Whitehelm tried to revive a sect of knights on his own to bring vengeance on the kingdom that locked us up

turns out my old group mates followed me to where a holy sword was buried that I was gonna use as both a BAMF weapon and bargaining chip. They got it in their heads I was %101 percent evil and slaughtered the knights and tried to kill me

he didn't win...they took the sword and killed all of his new comrades...the bastards let him live... that was their mistake

Whitehelm felt so betrayed at what happened he went on to revive the remnants of a shadow dragon into himself, turned an entire nation of darkelves into murder zealots, called on the dead mother of the most powerful witch in the land to counter her (pic related) introduced full zombie horde levels of necromancy to the setting, and started up an evil empire in the making with an ancient fire dragon to be his proxy

>playing coc
>one of my players wants to study an eldrich tome
>give him the look
>he ignores it
>2 sessions later he was a villanious npc from magic overuse

>Is it true this is the only fate for heroes that don't become martyrs?

There's a lot of career paths besides "I'm bad now."