Is there universe limit of acceleration? Or fastest acceleration that matter can handle?

That's a very good question. We don't really know. But we do know the Hubble constant appears to be stuck at roughly 160 km/sec per million-light-years. What it'll be, say, 1 billion years from now...no clue.

One thing. Space itself, which is what is accelerating, is not limited by the speed of light. In fact, many claim that space expanded faster than light in the first instances of the big bang when the universe expanded 10^30 to 10^100 times its original (big bang) size.

The cause of that expansion and acceleration might have been when a scalar field settled down to zero potential energy right after the bang. That extraordinary stress created...gasp...negative gravity, gravity that pushed rather than pulled. And, yep, the general theory of relativity allows for negative gravity in extreme cases.

See my source.