example of a function that could only be defined recursively is there a function that can be proved is only defineable recursively? the converse seems to be trivially false, i.e. every iteritive function is trivially defined recursively with 0 for coefficients for antecedal terms, how ever can a function such as $f(x)=x$ be defined non trvialy recursive in $\mathbb R$? $x_n = x_{n-1}+1$ defines it recursively over $\mathbb N$.
Is there a list of known ways that functions can be defined? for example consider followings:
$f(x)= \text{some algebraic expression , finite or infinite}$ : algebraicly defined
$x_{n+1}= \text{some algebraic expression involving }x_n, x_{n-1}, \cdots$ : recursively defined
$y : \text{if there are no 2 in decimal digits of x then 1 else 0} $ : defined by testing
or is there a reason not to bother with how functions defined?

I am not sure what you permit as non recursive function definitions, you seem to label them as iterative. That sounds like a loop. 
Now there are simple loops with fixed start and fixed end. Those can not
compute every computable function, they can deliver the primitive recursive functions only. 
This would be the case, where I would look for a function that can not be computed that way, it will have a recursive
definition which does. See here for examples.
If the loop parameters can change during the loop, or if a while loop is 
used then you get all computable functions. So you will not find an example.
This is the area of recursion theory or theory of computation.
There is a nice model, called register machines, which is one of the
simplest assembler languages possible, just adding, substraction and
conditional jump on test for zero. That is enough to compute everything
a Turing machine could do, only for natural numbers instead of words.
