I found one online that could:
https://keisan.casio.com/calculator
Apparently the answer is:
8.6958018964024964732812048202007848333...
24320437414562347129566458022240706625...
71418868460852622188800945E+99424
or ...
8.69762E+99424
I also found this:
The Golden Exponent? Any idea what this mysterious symbolism means?
https://www.deviantart.com/kthaven/art/The-Golden-Exponent-774399203?ga_submit_new=10%3A1543364489
NO! That is NOT the answer. No calculator can calculate it!! According to WolframAlpha, this is its representation as a power of 10s:
10^(10^(10^(10^(10^(10^5.842593328962333))))). However, they do give the last few digits as
.........1565172343.
And this online high precision calculator website:
https://keisan.casio.com/calculator
did calculate it.
Yes, they do give an answer, but it is totally wrong!!. The powers of 10 given by Wolfram are equivalent to the tower of 7s. You start from the right to the left like this: 7^7^7^7^7^823,543 =7^7^7^7^(3.75982352 E+695974). The power of the 4th 7 is more than 695,974-digit long!! And so on......etc.