What will the universe be like in a googolplex years time? It will be cold and dark in 10^(10^100) years. 

What else can be said of the state of things in the far distant future? Will everything have fallen into supermassive black holes and then radiated back into space so there is no matter left?

Will the universe revert to an exotic state as it was in primordial times, or will it slow and stay unchanged indefinitely? What kind of structure will it have?
That is nearly long enough to reach heat-death, which is estimated as about $10^{10^{120}}$. What that means is rather speculative, since it depends on various events that we have never observed, such as the spontaneous formation of black holes by quantum tunnelling. Such events are utterly rare, but are predicted to occur at very long timescales.

Any matter, will have at some time over a googolplex years quantum tunnelled into having enough mass to collapse as a black hole. So no matter will exist

Any black hole will have evaporated in Hawking radiation.  So no black holes exist.

Redshift will reduce the energy of photons to the point where their wavelengths are comparable to the scale of the universe.

Nothing is happening in the universe.

See https://en.wikipedia.org/wiki/Timeline_of_the_far_future

Now there is a: immovable object/unstoppable force paradox here, for in a universe that is spatially big enough, there are pockets of order among the chaos of heat death.