You can see that your constant is the solution to x=cos(x) (if the equality is true, you can replace the x on the right side with cos(x), and repeat this process).
The solution apparently has a name, Dottie constant [1].
Interestingly, I've found this kinda sorta occasionally works on functionals of a differential equation too if you start with a good enough guess. Someone has probably made this concept and the necessary/sufficient conditions rigorous but its getting into territory that is a good bit too advanced for me to follow.
As I recall from my fiddling, you are probably most likely to end up with a series solution of sorts, so it's a good idea to guess with polynomials or exponentials so you end up with component functions that form a basis for analytic functions.
It's not too bad to formalize :) The [https://en.m.wikipedia.org/wiki/Fixed-point_iteration](wikip... article on fixed point iteration) gives lip service to everything working fine in arbitrary metric spaces. It's easy to define a useful metric between functions (e.g. via integrating their difference, ignoring edge cases like occasional pointwise differences since a more careful treatment can give a presentation where those don't matter for the problem at hand), so the same kinds of theorems that work with cos(x) on the reals also work with differential operators on function spaces.
The solution apparently has a name, Dottie constant [1].
[1] https://mathworld.wolfram.com/DottieNumber.html