A colony ship with a Bussard Ramjet on each end (debunked theoretical spaceship drive that uses interstellar hydrogen for propulsion, i.e. the faster you go, the more medium you hit and the harder you accelerate) suffers an accident that destroys the deceleration engine. They keep accelerating because they need the engine’s magnetic field to protect them from interstellar dust. First they try reach the void between galaxies to safely repair the ship, when the interstellar medium is still too “dense,” they go for the void between galaxy clusters, then superclusters, then they just stay on the throttle until the big crunch, at which point, in the nothingness after the universe collapsed in on itself, they can finally fix their ship and begin decelerating into the bounce-back big bang of the forming next universe and colonize a planet.