Sequence when exposing

Hmmm... I'm sorry for not explaining the exponent towers earlier. I'll try my best to give a logical explanation on why exponent towers should be evaluated from the top downward despite it possibly being counterintuitive. To answer your original question, \(10^{10^{100}}=10^{\left(10^{100}\right)}\)

Let's use the variables a, b, and c and suppose that \(a^{b^{c}}=\left(a^{b}\right)^c=a^{bc}\), then the existence of exponent towers would be completely unnecessary as only one level of the exponent would ever be necessary; there must be a reason for such notation to exist. 

Another reason is that it is consistent with the order of operations. Exponents is a level below parentheses when it comes to priority, but the order of operations applys to the power, too. Here's an example:

\(2^{2+3}=2^5=32\)

\(3^{2*2}=3^4=81\)

And therefore, you must evaluate the exponent within the exponent. Does this make sense?

I learned today that even modern-day programs calculate have ambiguity in the matter. Excel 2013 is an example, as \(2^{3^2}\) outputs 64 as opposed to 256. In the future, 

X = (10^10)^100 = (10,000,000,000)^100 = 10^(10*100) = 10^1,000.

X = 10^(10^100) = 10^(1 + 100 zeroes) = 10 ^ 10 ^ 100 = 10^googol.

I think that when parenthesees aren't included, it makes the most sense to say....

a ^ b ^ c  =  a ^ ( b^c )

c  does not apply to  a  and  b  ,     c  applies only to  b  .