In Star Wars: Revenge of the Sith, it was revealed that Emperor Palpatine's face wasn't just the product of old age but also scarring from fighting the Jedi. Yet, in the galaxy of Star Wars, where missing limbs get replaced and wounds heal in Bacta Tanks, Palpatine never searched for a medical solution. Let's look at why the Sith Lord refused to fix his face for so many years.
When the Jedi attempted to arrest Palpatine, the Sith Lord didn't hold back as most of them fell within the first few seconds of the fight. But Mace Windu proved a formidable opponent, disarming and trapping Palpatine against a wall. The Sith Lord tried to use his signature move -- Force Lightning -- but it bounced back into him and began leaving horrible scars.
However, Palpatine chose to take the pain, as it left him appearing more injured and defenseless for Anakin's arrival. And his plan worked like a charm, with a wrinkled Palpatine begging for his life, making Windu out to be the villain. While it's unknown if Palpatine intentionally went into the fight wanting to become scarred, he wasted no time using it to his advantage with both Anakin and the entire Galaxy.
After returning to the Senate, Palpatine had to convince everyone that the Jedi were betraying the Republic and worth killing, and his new disfigured face was the perfect evidence. He even said as much, explaining, "The attempt on my life has left me scarred and deformed. But I assure you, my resolve has never been stronger!" That not only painted the Jedi as traitors but also garnered Palpatine enough sympathy to become Emperor.
However, once the Empire took control, it would've been easy for Palpatine to restore his face. But as someone who required power and everything he could possibly ask for, having good looks was the least of his worries. Palpatine chose to keep his face as a symbol of the Jedi's treachery and as a way to appear weak to his enemies.
The Sith Lord's rise to power based itself on appearing weaker than he actually was, and he continued that until his death. Even as Emperor, he wanted people's sympathy and to be underestimated, a tactic he used when confronting Luke in Star Wars: Return of the Jedi. Trying to turn Luke to the Dark Side, he purposefully left himself vulnerable, making the Jedi feel like he was striking a defenseless old man. So, while it was tough to look at, Palpatine saw advantages in keeping his appearance the way it was.