What this type of identities are called ? e.g. "expression containing no value/constant = value/constant" What are identities that on one side are free of any values, and just contain relationships/compositions between object/fucntion that do not contain any value at all?
For example from: What is interesting about $\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\sqrt{\log_x{\exp{\cdots}}}}}}}}}=\log_x{e}$?
it is remarkable because on one side there are no values, and on the other side there is value! what kind of sorcery is this?
On the other hand, equations like $\pi = \frac{A}{2r}$ is not a type of equation involving only relationships, as it contains relationship between 4 values/constants $ 2,\pi , A, r$.
I think I might have seen a relationship regarding graphs or networks that also had some type of similarity to equations relating non values/constants to a specific value/constant but I cant recall that at the moment. 
Is there a name for this type of identities? 
 A: If you interpret the expression on the left as starting on the left and then just keeps recursing through the $\log$'s and $\exp$'s indefinately without ever hitting a real number to stop the recursion then it does not make sense as a mathematical expression. A real-valued function needs some real-food to work on! 
It all boils down to: how is the left hand side defined? This is not really specified in the way the expression is defined, but the standard (only?) way to rigorously define such an expression is via a (finite) recursion of some form and then a limit. We pick a starting value $y_0$ and then calculate
$$y_{n+1}=\sqrt{\log_x\exp(y_n)}$$
If the limit $\lim_{n\to\infty} y_n = y$ exists then we can assign the value $y$ to the expression (and we should write $y(y_0)$ to be precise). In this case the value of the expression does depend on $y_0$. If $y_0=0$ then the limit is $y=0$ and if $y_0>0$ the limit is $y=\log_x e$.
There are examples of such recursions where $y$ is independent of $y_0$ like for example the recursion $y_{n+1} = \sin(y_n)$ then $y=0$ no matter what real value we use for $y_0$ so in this case I guess
$$\sin\sin\sin\sin\sin\cdots = 0$$
would make more sense, but remember that this is just notation - the real meaning is implicit. Bad notation is the root of all evil!
One class of expressions that quite naturally can have (or look like they have) no explicit numbers in them is operator identities. For example $\frac{d}{dx}$ is an operator that takes in a function and delivers it's derivative. Now $\frac{d}{dx}=5$ seems to be another example of an expression you are asking for. Again this does not make sense as a mathematical expression without some definitions of what it really means, but if we happend to work within the class of functions satisfying $\frac{df(x)}{dx} = 5f(x)$ then $\frac{d}{dx}=5$ could be used as a meaningful identity when manipulating formulas [$(\frac{d}{dx}+10)f=(5+10)f=15f$].
