Why is it so difficult to determine whether or not $\Large \pi^{\pi^{\pi^\pi}}$ is an integer?

That number has 666262452970848504 decimal places, so to determine if it's an integer you'd have to compute it with that precision. But this would take 270,000 TB, and we don't have many hard drives that large.

it is not difficult. The only problem seems to be is that it is to large number to present here all $10^{18}$ digits and the first digit after delimiter