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].
The solution apparently has a name, Dottie constant [1].
[1] https://en.wikipedia.org/wiki/Logistic_map
[2] https://en.wikipedia.org/wiki/Mandelbrot_set
[3] https://en.wikipedia.org/wiki/Attractor#Strange_attractor
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.