How many digits?

Number if digits in $8^{25}=\lfloor 25\log_{10}8\rfloor +1=\lfloor 5\log_{10}2\rfloor +1=\lfloor 75\times0.301\rfloor +1=\lfloor 22.575\rfloor +1=\boxed{23}$

Note $\lfloor .\rfloor$ represents the floor function

$8 ^ { 25 } = 2 ^ { 75 } = 32768 ^ { 5 } = 33778931862957161709568$ , so the numbers of didits of $33778931862957161709568$ is $\boxed { 23 }$ .

And $33778931862957161709568$ can be read as thirty-three sextillion seven hundred seventy-eight quintillion nine hundred thirty-one quadrillion eight hundred sixty-two trillion nine hundred fifty-seven billion one hundred sixty one million seven hundred nine thousand five hundred sixty eight.