Complexity class for polynomial algorithm for 3SAT, but with exponentially large values? I have a hypothetical question: suppose there exists an algorithm that solves an NP-Complete problem polynomial time, but requires the computation
of values that grow exponentially big (or small).  

For example, suppose POLY-3SAT solves 3-SAT in N^17, however to do so it must compute/evaluate the value of a number C whose value grows as N^1000 (or 1/N^1000).

What would this imply?  Does computing/evaluating an exponentially large value automatically place the algorithm in EXP-SPACE (or some other complexity class)?  It seems like this would be a different complexity class than just ordinary P.
Complexity classes are classes of problems, not classes of algorithms. For example, the existence of Dijkstra's algorithm means that the problem of computing shortest paths in graphs is in&nbsp;P, but it doesn't make sense to say that "Dijkstra's algorithm is in&nbsp;P".

Exponentially large values can be stored in binary in a polynomial numebr of bits, and we can compute on such numbers using a polynomial number of operations. This notwithstanding, it seems that you're trying to get a polnomial-time algorithm that somehow dows exponentially much work, which is an oxymoron.