I think you are confusing two different uses of the word "base". We say that our usual number system is "base 10" because "1232.3" means 1\times 10^3+ 2\times 10^2+ 3\times 10+ 2\times 10^0+ 3\times 10^{-1}. And "binary" is "base 2" because 1232.3 (base 2) means 1\times 2^3+ 2\times 2^2+ 3\times 2+ 2\times 2^0+ 3\times 2^{-1} which, in base 10, would be 8+ 8+ 6+ 2+ 1/2= 24.5.
But a number being the "base" of an exponential is very different. we can take any (positive) number as a base (I put 'positive' in parentheses because while, for many values of x, a negative number to the x power is perfectly well defined, there are some values of x such that a negative number or 0 to the x power is not defined). For example, for x= 2, \pi^2= 9.8696044010893586188344909998762...
And I think you may be misinterpreting "speculation". Of course, because \pi is an irrational number, it cannot be written as a finite number of decimal places and cannot be written as a fraction with integer numerator and denominator so I cannot write it or \pi^2 or \pi to any other power as a finite number or decimal places. I don't know what comes after that "09998762" that is indicated by the "...". I could theoretically use a calculator that holds a greater number of decimal places or use a computer program to extend to as many decimal places as I want but I would never get the entire value of \pi^2. But whether or not I can write it in a specific way, I know that \pi^2 is a specific number.
Similarly, although given a number a, I cannot actually calculate a_0, a_1, a_2, ... so that a= a_0\pi^0+ a_1\pi^1+ a_2\pi^2+ \cdot\cdot\cdot but I know that such number exist so that I can, in fact, write any number in "base \pi".
