A Time Machine has been set in motion before it was sufficiently tested... Its team of developers may end up being stuck in an endless loop cycle between the time it was first activated (4 minutes ago), the time it was first launched (now) and the time it can be relaunched (4 minutes into the future).
Time and player movement is kept track of on a board depicting one minute intervals between the first activation and the second launch. Gameplay involves players drawing and placing numbered/colored hexes. At the beginning of the game, before the first time machine launch, a line of hexes is laid out to create the sequence of events that led to the time machine being built. Then, after the first launch, each turn uses a new hex sequence where players try to recreate the sequence of events that created the first launch while also "fixing" it just enough to make it work correctly and hopefully break them all out of the loop. One aspect of the gameplay uses actual one minute time measures as part of the process.
Adding to the challenge, each time a player goes back in time, there is another one of them added to the game (depicting someone from the present traveling back in time and encountering themselves in the past). A light/dark color scheme is used to distinguish the present/past player tokens (dark blue/light blue, dark green/light green, etc.). Each new version of a player adds abilities and also causes potential problems, and there is a limit to the number of multiples a single player can have before the time line is destroyed beyond repair.
Play is cooperative, with a special commendation for the team member(s) who had the biggest part in fixing the machine. A game may also include a saboteur who is trying to subtly destroy the machine and prevent any progress in the field of time travel.