I came across https://github.com/philipl/pifs which is a fancy way of storing data. And a thought struck my mind, is it so that Pi is the simplest irrational number to calculate?
So the Question is. What irrational number is simplest to calculate eg. contains fewer/simpler operators?
Cost of operators are from low -> high (+-,*,/)
On
Firstly, I wouldn't call $\sqrt {-1}$ an irrational number.
To answer your question, recall that any rational number has either a finite or a repeating decimal expantion. So when you say 'easiest to calculate,' I might give you an irrational number exactly so that there is no calculation. For example, consider $0.1234567890112233...$, where it has each digit, then two copies of each digit, then 3 copies, etc. The fractional part (i.e. the part after the decimal) never repeats, so it's irrational. And there's no real 'calculation' involved.
This is all to say that your question doesn't admit a rigorous answer.
First, we already know that "almost all" of the real numbers are normal. Second, despite the first fact, it is quite difficult to prove a given number as normal or non-normal. Third, a number may be normal in one base but not in another. We don't know if $\pi$ is normal in base 16 and we don't know if $\pi$ is normal in base 10. Learn more about normal numbers here.
Further, irrational doesn't automatically imply normal. Liouville numbers are irrational but not normal because for example they contain way more zeros than any other digit. A normal must have a uniform distribution of all the digits in its expansion.
Lastly, what about something like Champernowne constant which can be obtained by just adding one and concatenation. And this number is proven to be normal. And the proof is easy. Every number (and hence every file in base 10) is in there because every number is in there. And not just that but we also know exactly where a given number is. You are probably looking for a normal number in base 2 or base 16, not in base 10.