With the expression 62, we have already far surpassed the googolplex! The result of 2(265536) is a number of about 6.031226*1019727 digits. One way to think of this number is: Take 2, square it once, and you get 4. Then square your number again, and you will end up with 16. Then continue by squaring the number 65534 times! Even if every person in the world had a computer with a 1000-terabyte hard drive, then you would only have 7.5*1024 bytes, which is nowhere near what is needed to even begin to write the decimal expansion of this number.
It turns out that the leading digits of this number can be computed, by taking the log10(2) accurate to at least 19735 decimal places, and multiplying it by the decimal expansion of 52. Below are the first and last 40 digits of 62.