Infinitely valued functions Is it possible to define a multiple integral or multiple sums to infinite order ? Something like $\int\int\int\int\cdots$ where there are infinite number of integrals or $\sum\sum\sum\sum\cdots$ . Does infinite valued functions exist (Something like $R^\infty \rightarrow R^n$ ) ?

Yes, it is possible to define multiple integrals or sums to infinite order:
here is my definition: 
for every function $f$ let 
$$\int\int\int\cdots \int f:=1$$ 
 and
$$\sum\sum\cdots\sum f:=1.$$
As you can see, I defined those objects.   
But OK, I understand that you are looking for some definitions granting some usual properties of the integral. Here is another answer: 
it is possible to define integrals of functions between Banach spaces. There are measures on infinite dimensional Banach spaces (for example Gaussian measures) so this might be the concept which is "meaningful" for you. For example you can consider a Gaussian measure on the space of continuous functions $C([0,1])$ induced by a Wiener process and you can calculate integrals with respect to that measure. With some mental gymnastics you can think about those measures and integrals in a way you asked about. 
