This kind of thing happens even if you don't assume time repeats. As far as we can measure, the universe is spacially infinite. Finite age, infinite spacial extent. If you ever hear someone talk about the diameter of the universe, they're talking about the diameter of the observable universe - the part of the universe close enough for its light to actually reach us. But as far as we can measure (based on measurements of large scale spacial curvature), the universe is truly, literally infinite. It's possible it curves back on itself with a radius much larger than the diameter of the visible universe. But if it truly is infinite, infinities make some very weird things possible.

For example, a diameter the size of the observable universe has a finite number of states. There are only so many atoms and so many ways to arrange those atoms. This number is unfathomably large, but it's not infinite. But in an infinite universe, on a large enough distance, everything repeats. It doesn't repeat in a regular pattern, but it does repeat. So get in a space ship and fly off into space. If you could go far enough, eventually you would run into an exact duplicate of yourself, with all the memories and life experiences you have - an atom-for-atom copy of yourself. Travel far enough and you'll encounter a duplicate of our entire observable universe. And worse still, if the universe truly is infinite, there must be an infinite number of such copies.

I'm not talking about alternate realities or parallel universes here. I'm not talking about getting in a time machine and visiting an alternate timeline. I'm not talking about the quantum many worlds theory. I'm talking about the very space you inhabit, this universe. If you go out in a space ship and could travel arbitrarily far, it would eventually seem like you had come right back home, even though you're quintillions of light years from where you were born.

Infinity is a terrifying thing.