If a uniquness for all functions exist shouldn't there be uniquness to recursion? What I'm specifically saying is every function is definitely unique, as they may be nearly equivalent to another function, for example.
Let's make a table of values for $^{x}2$
(0,1)
(1,2)
(2,4)
(3,16)
(4,65536)
The recursive relation is of course, $a_2=2^{a_1}$, if $^{x}2$ is described as this recursively is it possible to describe these points in others ways that don't simplify to $a_2=2^{a_1}$
This may sound obvious, but if every function is unique analytically, than is a recursive function unique.
So does $^{x}2$ have a unique recursive function, or are there other ways of describing it recursively,without simplifiying it to $a_2=2^{a_1}$? 
I'm having trouble explaining this, hopefully this could be clarified!

Any relation that produces this sequence can of course be 'ultimately' simplified to  $a_{n+1}=2^{a_n}$ just because this is how consecutive elements in the sequence are related. You do not however specify which identities are 'allowed' in a simplification. There are analytic functions that vanish on all integers, $\sin(\pi x)$ for example, so $a_{n+1}=2^{a_n}+\sin(\pi a_n)$ will produce the same sequence because all $a_n$ are integers, but as functions $2^x$ and $2^x+\sin(\pi x)$ do not reduce to each other. Also, 
in general there is no algorithm to determine if two analytic expressions represent the same function, or can be reduced to each other based on a given list of identities.
