The problem is not with infinity per se.

If the universe gets cloned one time every second that would still be a huge claim.

A million times every nanosecond is even a bigger claim.

Now imagine how many states the universe can be in, and how many times a second can be meaningfully divided.

If you can solve the "one clone every second" then I would be satisfied.

What is the problem, though? Who says the uni/multiverse has a finite memory capacity? For more in this vein: http://www.preposterousuniverse.com/blog/2014/06/30/why-the-...

It's not about the memory capacity ..

What is it about?

I'm not reading any problem description in your comment other than an inadequate imagination.

Note that you're assuming that a nanosecond is a tiny amount of time. It's tiny relative to what we're used to, but as far as what "implements the universe" goes, a nanosecond could be enough time for a huge amount of occurrences to happen within. We don't know.

I did not assume nor imply it's tiny in any "absolute" sense.

Then why is "every second" a "huge claim" and "every nanosecond" "even a bigger claim"?

It's only linearly bigger I suppose.