Darth Sidious fears Luke Skywalker in the same way he feared Anakin before he managed to manipulate him into becoming his apprentice. He fears how powerful he can become. When Yoda fought Darth Sidious and then escaped, why couldn't Palpatine sense his presence?
As powerful as Darth Sidious may have been, it's easy to forget he actually seemed afraid of Master Yoda.
Brand reveals that Palpatine isn't entirely composed of hate, but also of fear, and in comforting the dying Brand, Luke Skywalker explains, "More than anything, the Emperor fears obliteration in the dark side." For all his posturing, Palpatine still fears death and the consequences of his use of the dark side.
Emperor Palpatine admitted to fearing Luke Skywalker and even said that if he had him by his side instead of Kylo Ren, he would have won the galaxy.
Jedi Master Yoda is the most powerful of them all. This should come as no surprise, as he was the head of the Jedi Order in the prequels and was revered for his abilities during the original trilogy. Yoda's power is unmatched and was a real threat to Darth Vader and the Imperials.
Thus, while Qui-Gon was less powerful than Yoda, Palpatine feared him more.
Palpatine absolutely did NOT fear Kenobi as a physical threat to himself. Could he have beaten Obi-Wan in a straight lightsaber duel? Possibly. Let's compare their forms.
15 Darth Plagueis
Not only did he survive multiple attempts on his life, he also brutally destroyed his Sith Master. Without Plagueis, there is no Sidious — and without Sidious, there is no Darth Vader.
Even after his brutal scorching in the fires of Mustafar, sand remained the only natural phenomenon that Vader feared. To him, sand was a reminder that he'd failed to save his mother or Padme from death.
Here, Rey is directly facing her fear, her ultimate worry that she may never know where she comes from or who she truly is. Rey asks herself what she must endure to find herself. It is loneliness, isolation, scatteredness.
Yoda still stands as one of the most powerful characters in Star Wars, yet even he couldn't defeat Palpatine. After their battle in Revenge of the Sith, Yoda felt beaten and fled into exile on Dagobah, where he hid away for 19 years.
Darth Sidious was able to cheat his upcoming death by transferring his spirit into a clone body while falling into the reactor core of the second Death Star. He later sought to prolong his existence by futilely manipulating his grandchild Rey into killing him so he could live further by possessing her body.
Amedda was one of the few beings in the galaxy who knew Palpatine's secrets, most notably his true identity as a Sith Lord.
Dooku voluntarily left the light side behind and became Darth Sidious' dark side disciple, taking the secret name Darth Tyranus and leading the Separatist army. Dooku advanced Sidious's secret plot to take over the galaxy, but forgot that betrayal is the nature of the Sith.
Darth Plageuis is one of the most famous Sith Lords in the Star Wars Universe. Plageuis was the mentor and master to Darth Sidious, and he was spoken of to Anakin in Star Wars: Revenge of the Sith. Plageuis was unique as a Sith Lord because he had the ability to prevent death and create life.
Palpatine wouldn't have feared Anakin at the present moment (he was still not as powerful), but Anakin was always talking about 'overthrowing' the Emperor. It even showed in a PS2 game (forgot the name) that Anakin would have killed Palpatine if he had won against Obi-Wan on Mustafar.
Infil'a put up a fierce fight against Vader, almost killing him by throwing the already damaged Sith Lord off of a cliff. Despite severe damage to his suit, Vader survived the fall, and repaired himself with parts from Infil'a's droid, Arex.
1 Darth Sidious
There has been no Sith, as powerful, meticulous, nor as evil as the Emperor himself, Darth Sidious. As previously mentioned, Sidious trained in the Dark Side by Darth Plagueis before Sidious killed Plagueis in his sleep, making Sidious a Sith Master.
But the record Sith-killer is actually Darth Bane himself. Darth Bane, founder of the Rule of Two, killed more Sith than any Jedi in the history of the Star Wars galaxy. Thousands of years ago, a schism among the Jedi Order led to the creation of a new faction - one devoted to the dark side of the Force.
Sidious himself felt that he and his peers had outgrown the use of lightsabers, and only continued carrying them to mock the Jedi. He ultimately viewed his own lightsabers as little more than an affectation, and rarely wielded them in combat.
He was still completely unaware of the Sidious/ Palpatine connection until ROTS. He also said during TCW season 6 that the Jedi order must believe in the clones and end the war before the Sith's plans are realized.
It's implied that this isn't feigning and Palpatine is actually fighting to win. In the book its implied that he allows Mace to move him and make certain moves, however this is a not uncommon dueling tactic, waiting for openings and surprising an opponent.