I am aware that Catalan's constant appears in the evaluation of many definite integrals, as well as in the evaluation of certain infinite series, and is a special value of a function closely related to the Riemann zeta function, and so on.
But is there a way in which we might think of this constant as important in its own right, and not come at it 'indirectly;' not approaching it only through other 'loftier' concepts first, or using it as a convenient shorthand for the numerical result of a certain integration or summation process, etc., but through some simple concepts that first lead you to the idea of some important constant, whose more sophisticated properties we can then deduce, and in doing so derive its relations to the transcendental functions, integral and series representations, so that we arrive at the idea of Catalan's constant itself as some important concept in its own right, without appealing to it as kind of a secondary idea?
For example, we can do this with $\pi$ and $e$ quite easily. The concept of $\pi$ appears as a simple geometric idea. I only need to understand some basic geometrical ideas, and then it would be only a matter of time before I arrived at the idea of $\pi$ and understood its importance, even if I didn't fully understand the full depth of its importance. The same with $e,$ or even the Euler-Mascheroni constant $\gamma.$ I only need some basic concepts of calculus.
With more sophisticated concepts I obviously need more sophisticated prerequisites, but regardless, it is often the case that many fundamental ideas actually have a very simple, and very intuitive germ, which is why you see them pop up again and again, and often there is still some fairly simple relationship between ideas which tells you immediately that something may be important, even if those ideas are themselves not quite 'basic'.
With Catalan's constant, I haven't seen any kind of explanation that explains this. It's almost always 'Catalan's constant appears in the evaluation of ...' which only tells you that it is important, but not really where this importance comes from, or why you would be led to care about this constant as anything more than a useful notation for some integral or sum, unlike many other ubiquitous objects in mathematics, where you might first grasp them as an important idea, and then notice their many other uses.
Can anyone explain why Catalan's constant is important, or why you would be motivated to care about this constant in its own right? I appreciate that this question may be vague but I'd love to hear any good answers.