Perhaps--

The purpose of life is to experience the finite.

I think it was Alan Watts who said something like we are God playing hide and seek with ourself.

Note how video games have the same appeal. Lives and deaths and the point being to live as long as possible, then die, then do it all over again. We also do this in movies and for many thousands of years through drama.

So infinity, and by that we mean an infinite time-space environment at the very least (and probably realms far beyond what we're aware of), there would be forms of consciousness that would have evolved enough to survive death, whether biological or by simulation. But they would then look for stimulation, right? Entertainment. How else to entertain oneself but by creating a finite existence.

And wouldn't this be the fall from grace? The forbidden fruit? The dive into a finite existence? One full of pleasure and pain.

And of course it will be endless, a fractal of sorts but one that is infinitely unique with every iteration, because with infinity comes the possibility for infinite uniqueness. So indeed uniqueness, newness, is the hallmark of infinity. It's mindfuck of sorts... I previously viewed infinity as being a state in which every potentiality had been realized and is being realized an infinite number of times, but the 'of time's' element is a finite mapping onto the concept, and that doesn't work.

I now realize that infinity would be defined by nothing. It would have no definition. Newness, uniqueness.... that is infinite. The moment.