Complexity class associated with exhaustive search
It's a little vague, but I like the question. I wrote a paper about it a LONG time ago. 
Maybe this will help the Anonymous questioner: 

Brute Force Search and Oracle-Based Computation 

Here's a summary. Informally, if you do not keep any scratch work from previous trials, and just try all possible solutions in lexicographical order until a desired solution is found, then brute force corresponds precisely to $P^{NP}$. If you keep around even $3$ bits of scratch work from one possible solution to the next, then you can do $PSPACE$, via Barrington's theorem. There are other possibilities, such as what happens when you don't run in lex order but according to some other efficiently computable list of all strings.

[WARNING WARNING WARNING: It's likely that I disagree with almost all opinions stated in this paper. It was written about 10 years ago, by someone with the same name but who is essentially a different person.]