Time travel is a fiendishly complicated subject, especially when the notion of time loops come into play. For this reason, Deathloop, and other video games with time travel gameplay, are impressive achievements, the product of dedicated developers who had to overcome formidable technical and design challenges.
Most people learned about the concept of time loops from the movie Groundhog Day, the 1990's comedy where a weatherman (played by Bill Murray) becomes trapped in time, living the same day in a small town over and over. The success of Groundhog Day wound up inspiring a whole sub-genre of time-loop dramas: Edge of Tomorrow, starring Tom Cruise and Emily Blunt, was about time-looping soldiers who must discover the perfect strategy to halt an alien invasion.
Interestingly, All You Need Is Kill, the Japanese novel adapted into Edge of Tomorrow, was inspired by video games – specifically, video games that let a player manually save and reload a specific game state. In effect, any video game with the ability to save your progress features time loops as part of their gameplay, if not their story. The following video games interesting because they take the "repetition of challenging experiences" core to video games and integrate them into a story of time travel, resets, and tough choices.
Undertale's Save and Load Mechanics
In most RPGs, the ability to save and load games is a break in immersion. In Undertale, the famous RPG "Where No One Has to Die," the ability to Save and Load is the player character's unique special ability, a power that they can use in-universe to replay game events over again and choose the game ending they desire, whether said game ending involves "Saving" everyone or killing them all. A few key characters in Undertale are even aware of the main character's time-loops, reacting (ominously) to the player whenever they reload the game or make decisions based on future knowledge.
12 Minutes' Murder Mystery
In 12 Minutes, an minimalistic indie video game published by Annapurna Interactive, the player takes on the role of a husband in a small apartment who gets brutally murdered by a police officer who breaks in and accuses his pregnant wife of committing murder. After each death, the player wakes up 12 minutes in the past, forcing them to experiment with new approaches and gather the information they need to not just break free of the time loop, but to also unravel the mystery behind the murderous police officer and the secrets his wife carries.
Titanfall 2's Best Level
The single-player Titanfall 2 is filled with many brilliant levels. One of the best, "Effect and Cause," has Pilot Jack Cooper and Titan chassis BT-7274 stumble across a ruined research facility armed with a special glove that lets the wearer travel through time. The bulk of this level features Jack time-hopping between the present-day ruined facility and the pristine past facility at will, using the gauntlet's function to escape enemy attacks, by obstacles, and eventually become the catalyst for the facility's foretold destruction.
Elsinore's Hamlet Loops
The point and click adventure game Elsinore takes the "time-loop triggered by death" mechanic and applies it to the famous Shakespeare tragedy Hamlet. The protagonist of Elsinore is Ophelia, the forlorn love interest of Hamlet, who finds herself living the same four days of tragedy over and over against, searching for a timeline in which her loved ones don't have to kill each other... and where she doesn't have to listen to Hamlet's teenage brooding.
Deathloop
An spiritual successor to the Dishonored stealth game franchise by Arkane Studios, Deathloop's time-looping protagonist is an assassin named Colt, trapped on the 1970s-styled island of Blackreef, whose decadent inhabitants seem to be engaged in a Purge-style murder fest. To escape his time loop, Colt must assassinate 8 targets in a single night, but a rival assassin named Julianna has been hired to get in his way. In an interesting twist, Arkane Studios is developing a multiplayer mode where players can invade each other's worlds, thus pitting two time-looping killers head to head.
There are lots of other games featuring time loop mechanics (such as 2019's Outer Wilds) but the ones mentioned above are some of the best and most unique examples around.