Is it possible to calculate the leading decimal digits of $3 \uparrow\uparrow 5\ = 3^{3^{3^{3^3}}} = 3^{3^{7,625,597,484,987}}$? Using currently known methods, this would require knowing the complete decimal expansion of $3^{7,625,597,484,987}$ (which has $3 638 334 640 025$ digits), and the common logarithm of 3 up to at least that many decimal places.
While this would not be possible on an ordinary desktop computer (which maxes out at billions of digits), it may be doable with more effort. However, as far as I know, no one has tried to push these calculations to the limit in the same way as has been done for calculating $\pi$ and $e$.
As I mentioned in another recent post, this is in fact my goal if I do find a better way to calculate the leading digits of very large powers.