Starting off our list with an anime that is a stone cold classic, even if it kinda stumbles adaptation wise near the end there.
Erased is the story of a man named Satoru, an unassuming young man save for one ability: the ability to travel back random amounts of time whenever something horrible happens, usually just barely enough time to avert the disaster. When his mother is suddenly murdered one night, Satoru is transported back to when he was 10 years old, around the time unsolved serial murders EERILY similar to how his mother was killed were taking place. Satoru realizes that he must solve these murders and catch the one responsible if he has any hope of returning to his time and saving his mother. He also has to do this while being a 10 year old.
Everything but the last episode or two are 100% faithful to the source material, but the ending was so bad that it dragged it down horribly for fans and right into this list. There was a live action show made after the book series was finished that has the original ending, but people still prefer the anime version for its much more creative direction - as opposed to the flat, boring look of the live action show. If you know anything about color theory, the anime is VERY good at giving you the tools needed to figure it all out for yourself.
Still, if you can stomach an unfaithful ending, Erased is the kind of mind bending thriller mystery that'll keep you hooked.