Naruto could be often involved. Since he's Kakashi's student, he might be part of the events that keep repeating in the time loop. Kakashi could be trying to protect Naruto in different ways each time the loop occurs.
Sasuke is also likely to be a main character in such fanfiction. Given his complex relationship with Kakashi and his own importance in the 'Naruto' series, his actions could be a key factor in the time loop. Maybe Kakashi is trying to prevent Sasuke from leaving the village in the different loops.
Another character could be Obito. Since Obito has a deep connection with Kakashi's past and his development as a character, in the time loop fanfiction, Obito's presence could be used to explore Kakashi's guilt, redemption, or changes in their relationship. For example, Kakashi might be trying to change the past events related to Obito within the time loop.
Sansa Stark is frequently in these fanfictions. As someone who was a victim of Joffrey's cruelty, if he is in a time loop, Sansa might find herself in a different situation each time. Maybe in one loop Joffrey starts to treat her better, which would completely change her perspective and her own actions within the story. Also, Cersei Lannister could be involved. As Joffrey's mother, she would be very concerned about any changes in her son during the time loop and might try to interfere to keep things as she wants them.
Often, Gai is paired with Kakashi in Hurt Kakashi fanfiction. Gai, with his unwavering belief in friendship and his eternal rivalry with Kakashi, can be a great character to interact with a hurt Kakashi. He might try to pull Kakashi out of his slump through their friendly competition. Also, Rin is sometimes included. Since she has a significant past with Kakashi, her presence in the fanfiction can either be a source of comfort or more pain, depending on how the story is written.
Sasuke could be a popular character. He might be involved in the story in a way that either helps or hinders the situation where Naruto hurts Kakashi. For example, if Sasuke tries to stop Naruto from doing something stupid that would hurt Kakashi.
In most 'time loop Harry Potter fanfiction', Harry is front and center. His character development within the loop is usually the focus. But Hermione is a key figure too. She could be the one to discover the magical theory behind the time loop. For example, she might find a long - forgotten spell or potion that could be related to time manipulation. Ron, although sometimes overshadowed by Harry and Hermione, can bring in the element of friendship and loyalty. In some cases, characters from the Death Eaters side like Bellatrix Lestrange might be involved if the time loop is related to a major battle or a plot against Voldemort. Their presence can add more danger and complexity to the story.
Liara is another character that frequently shows up. Her knowledge of Prothean technology and history might be crucial in understanding and potentially stopping the time loop. She could be using her skills as a scientist and her connection to Shepard to figure out what's going on.
Perhaps in the 'kakashi time loop fanfiction', Kakashi is forced to relive a day that has a great significance in his past, like the day his father died. Through the loop, he gets the chance to come to terms with his feelings and past mistakes in a way he couldn't before. This could be a really emotional and character - developing aspect of the fanfiction.
Iruka could be included. He was Naruto's first teacher and has his own connection to Kakashi. His presence can bring out the different teaching styles between him and Kakashi. Another good character is Jiraiya. His relationship with Naruto as his godfather and his connection to Kakashi through being comrades can add some interesting storylines.
Sasuke could be a great addition. He could be a rival to Naruto even at a young age, and Kakashi would have to deal with their constant bickering. His brooding nature would add an interesting dynamic to the story.
It really depends on the source material. For example, if it's based on a superhero franchise, it could be the main superhero like Superman or Batman. If it's a fantasy - based one, perhaps a wizard or a chosen hero from the original work. There's no one - size - fits - all answer as different fanfictions can be based on different universes and thus feature different popular characters.
Naruto would likely be involved. He has a strong bond with Kakashi and would surely try to rescue him. Sasuke could also be part of the story. Given his relationship with Itachi, it would add an interesting dynamic.