It's super common for creators to base anime on manga or light novels, but anime based on novels are considerably more rare. Even so, full-length novels can serve as excellent source material for anime. Literary fiction is usually lacking in ridiculous anime tropes, so adapting them can be a great way to break the mold with some fresh ideas. They can be hot takes on classic novels like The Count of Monte Cristo by Alexandre Dumas, or they can be fun reworks of modern books, like Paprika by Yasutaka Tsutsui.
Anime inspired by novels can add variety to your anime viewing experience. Plus, anyone who's seen the musical will undoubtedly be morbidly curious about what a shojo adaptation of Les Misérables entails.
Hayao Miyazaki's film Howl's Moving Castle is based on a once-obscure fantasy novel by British author Diana Wynne Jones. The book was first published in 1986, and initially found modest success. However, after the movie came out, the novel won the Phoenix Award in 2006, which recognizes older childrens books that were overlooked when they were first released. If you can't get enough of Howl being extra, the book is well worth checking out.
#57 on The Best Animated Films Ever
Hyōka is based on a series of young adult mystery novels by Honobu Yonezawa, called The Classics Club Series. Not only has the story received an anime adaptation, it's also been turned into a live-action movie and a manga.
If you're looking for a faithful adaptation of The Count of Monte Cristo by Alexandre Dumas, Gankutsuou: The Count of Monte Cristo is probably not the right series for you. While it sticks relatively closely to the plot of the novel for the first 18 episodes, the anime goes off in a completely different direction after that. Notably, the setting shifts several thousand years into the future, and onto another planet.
Based on a novel of the same title by Tatsuhiko Takimoto, Welcome To The NHK is a series that most anime fans have at least a passing familiarity with. It's the story of a 20-something shut-in's struggle to lead a normal life, and it is equal parts hilarious and heartbreaking. One major difference between the book and the anime is that in the series, the main character's vivid hallucinations appear to be the result of mental illness, whereas in the novel, he's explicitly using hallucinogenic drugs.