Question

Does the universe ever stop ?

Answer 1

We would only know that if we could see it.

So far, with the most sensitive astronomical instruments that have
been built, we don't see an end as far away as we can see. 

The farthest (dimmest) objects we can see are something like
13-1/2 billion light years away.

Answer 2

Nope. But nobody knows for sure. They haven't gone that far out. It could take billions of lightyears.