As phrased, it is misleading.

Are we talking about until the end of the universe? (Which IS a finite amount of time.) The current theory is that the universe as a whole will finally fizzle into decomposed hydrogen in about one trillion years.

Or are we talking about eternity? (Which by definition is an infinite amount of time.)

Imagine a googleplex number of "years" (-years are a human construct based on earth's revolving around the sun). Multiply that googleplex number of years up to the googleplexth power. Take that resulting number of 'years' and multiply that again up to the googleplexth power. Repeat that same process another googleplex to the googleplexth power number of times. And repeat that again by a googleplex to the googleplexth number of years. Etc...

Eventually the chimpanzee would succeed. The universe would be LONG GONE by then and the only thing that might exist would be a chimpanzee and his type writer. But eternity never ends.