I have noticed that among the many mathematical constants (particularly irrational or transcendental), some require more comprehensive equations and descriptions than others, in order to specify their value to arbitrary precision. This is a difficult idea to put into words concretely, but I will do my best in the following paragraphs, as well as mark this as a soft question.
This idea came from thinking about how much "information" is within the digits (or bits) of $\pi$, for example, from a computational perspective. Initially, one might note that it requires countably infinite amount of memory to store all the bits of $\pi$. But this is rather inefficient, as we could instead simply store an equation for $\pi$, like the famous $$\pi=4\sum_{n=0}^\infty\frac{(-1)^n}{2n+1},$$ which requires only a finite amount of symbols. But, this leaves us with questions like
How can we measure how much "information" is in this equation / the symbols within the equation?
If there does exist a method to measure the information, what is the minimal equation (or algorithm) which could be used in theory to calculate $\pi$ to any precision?
Furthermore, we could instead store the English sentence $$\text{The ratio of the circumference to the diameter of a circle,}$$
which also encapsulates enough information to specify the constant $\pi$, albeit with subjectivity and overhead via the assumption that the reader understands English & basic geometry terminology.
Lastly, one last idea would be to compare the file-size of computer programs which calculate the digits of $\pi$, but this too certainly depends on things like which language we are using.
Furthermore, we could contrast these ideas with another constant, like the Euler-Mascheroni constant, $\gamma$ which arguably requires more complicated equations / descriptions / algorithms to specify. So,
Are there objective mathematical measurements of how much "information" is required to construct given (real) numbers?
I assume my questions border many areas of math and computer science, particularly Kolmogorov complexity, Information theory, and the Arithmetical hierarchy, but I lack the experience with this sort of idea, so any and all insight would be greatly appreciated.