Finding anime that doesn't have fan service is challenging, to put it mildly. Sometimes it feels like it's in everything, not just in genres like ecchi that are supposed to be fan service heavy. This is a problem. Not everyone likes fanservice or wants to see it all the time. For some, it can feel demeaning, sexist, or even predatory, ruining their experience of an otherwise excellent show. Even people who enjoy fan service don't necessarily want to see it when they are, for example, watching anime at their parent's house or on the bus.
Luckily, there are plenty of great anime without fan service out there, even if it doesn't always feel like that's the case. From slice of life to action adventure, there are anime in every genre that leaves out the T&A and focuses on telling a good story.
As a child, Shouya Ishida bullied his deaf classmate Shouko Nishimiya. In his teen years, he realizes that what he did was wrong, and he tries to make up for it. This beautiful movie is deeply in touch with its characters' emotional lives and doesn't need fan service to carry it.
After accidentally insulting the Witch of the Waste and being cursed to rapidly age into a 90-year-old woman, she seeks out help from the bubbly and bizarre wizard Howl. Like the rest of Studio Ghibli's filmography, Howl's Moving Castle doesn't deal in fan service.
Cells At Work features anthropomorphized cells working hard to maintain the human body that they all call home. It's so safe to watch in public that teachers have been using it to educate kids.
Ginko is a mushi-shi - a person whose job it is to solve the problems caused by mushi, mysterious lifeforms that can grant psychic powers, trap people in never-ending forests, and more. Fan service would likely detract from Mushishi's atmospheric and quiet tone.