I have heard that it is unknown whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer. How can this be? $\pi$ is known to many digits and it seems like only a matter of time on a computer to find the integer part or to find that there must be a decimal part to the number.
What makes it difficult to determine? I know that it is overwhelmingly likely that the constant is irrational but I am interested in why it is hard to show.
Another aspect of the question could involve answering why there is no proof that for $\alpha$ transcendental $\Large \alpha^{\alpha^{\alpha^\alpha}}$ is not an integer. I am not naive enough to think such a proof would be easy but I also don't know too much about it to know why it would be hard.