If you take the gamma function of a real positive number n, infinitely recursively (gamma(gamma(gamma..(n))).., for various values it either goes down to 1 or goes to infinity above a certain value. The first few digits of the value where it would just evaluate back to itself seem to be around 3.5623824.
Questions
Does this transcendental number already have a name?
Is it equivalent to any differently rendered analytical solution related to anything else for example the Golden Mean?
Does it have a real-world engineering application?
As @A rural reader commented, you are looking for the non-trivial zero of function $$f(x)=\Gamma(x)-x$$ Using numerical methods does not make any problem. To have something nice, knowing that the solution is close to $x=\frac 72$, we can use one single iteration of high-order iterative methods and have explicit approximations.
For Newton method $$x_{(2)}=\frac 72+\frac{28-15 \sqrt{\pi }}{15 \sqrt{\pi } \psi \left(\frac{7}{2}\right)-8}=3.566255560\cdots$$
We can have this kind of expression for any order but, even for Halley, the formula does not fit on a single line. So, I shall just give the numerical values as a function of the order of the method.
$$\left( \begin{array}{ccc} n & x_{(n)} & \text{method} \\ 2 & 3.5662556 & \text{Newton} \\ 3 & 3.5622765 & \text{Halley} \\ 4 & 3.5623845 & \text{Householder} \\ 5 & 3.5623822 & \text{no name} \\ 6 & 3.5623823 & \text{no name} \end{array} \right)$$
This number has not been identified by inverse symbolic calculators.