If the universe is expanding indefinitely, then it means it has unlimited potential to store information, for example as the position of matter inside the universe.
For example imagine a spacecraft going away from earth, with a mirror toward the earth. On earth you can send a stream of bits with a laser toward it and it will come back. As it goes farther and farther away, you can increase the amount of bits in transit in the stream, making it a way to store a potentially infinite amount of bits.
Of course this is not a really good example, because I'm sure there is a lots of reasons why this scheme would not work indefinitely, such as chaos theory preventing us to target the spacecraft with the every-growing precision required.
But still, I would love to see an actual reasoning of wether or not the information we can store in the universe is actually bounded.