Another hint if you want to do a quick calculation (without calculator) for getting an approximation (of base 2 numbers):
2^10 ~ 10^3 (1024 vs 1000)
Now you can convert e.g.
2^200000 into 10^60000 (divide by 10, multiply by 3)
If you have base 10, the exponent gives you the number of zeros (or, the number of digits).
If you have something like 357 * xxxx then it has 2-3 digits more (as many digits as the number - here 357 - has). The 2/3 depends on whether it just 'flips' to the next digit or not. (I might be inaccurate here - anyway, it's only about few digits so doesn't really count on the large scale).
Coming now to the Sophies, and give an example here (which is currently on the homepage):
2^666666 ~ 10^200000
+ 14 digits from the multiplier
So you see, I'm a bit off here, but it gives you the right scale quickly.