What is the complexity class most closely associated with what the human mind can accomplish quickly?
I don't claim this is a complete answer, but here are some thoughts that are hopefully along the lines of what you're looking for.

NP roughly corresponds to "puzzles" (viz. the NP-completeness of Sudoku, Minesweeper, Free Cell, etc., when these puzzles are suitably generalized to allow $n \to \infty$). PSPACE corresponds to "2-player games" (viz. the PSPACE-completeness of chess, go, etc.). This is not news.

People generally seem to do alright with finite instances of NP-complete puzzles, and yet find them non-trivial enough to be entertaining. The finite instances of PSPACE-complete games that we play are considered some of the more difficult intellectual tasks of this type. This at least suggests that PSPACE is "hitting the upper limits" of our abilities. (Yet our opponents in these PSPACE-complete games are generally other people. Even when the  opponents are computers, the computers aren't perfect opponents. This heads towards the question of the power of interactive proofs when the players are computationally limited. There is also the technicality that some generalizations of these games are EXP-complete instead of PSPACE-complete.) 

To an extent, the problem sizes that arise in actual puzzles/games have been calibrated to our abilities. 4x4 Sudoku would be too easy, hence boring. 16x16 Sudoku would take too much time (not more than the lifetime of the universe, but more than people are generally willing to sit to solve a Sudoku puzzle). 9x9 seems to be the "Goldilocks" size for people solving Sudoku. Similarly, playing Free Cell with a deck of 4 suits of 13 cards each and 4 free cells seems to be about the right difficulty to be solvable yet challenging for most people. (On the other hand, one of the smartest people I know is able to solve Free Cell games as though she were just counting natural numbers "1,2,3,4,...") Similarly for the size of Go and Chess boards.

Have you ever tried to compute a 6x6 permanent by hand?

I suppose the point is that if you take natural problems in classes significantly above PSPACE (or EXP), then the only finite instances that people are capable of solving seem to be so small as to be un-interesting. Part of the reason "natural" is necessary here is that one can take a natural problem, then "unnaturally" modify all instances of size $&lt; 10^{10}$ so that for all instances a human would ever try the problem becomes totally intractible, regardless of its asymptotic complexity.

Conversely, for problems in EXP, any problem size below the "heel of the exponential" has a chance of being solvable by most people in reasonable amounts of time. 

As to the rest of PH, there aren't many (any?) natural games people play with a fixed number of rounds. This is also somehow related to the fact that we don't know of many natural problems complete for levels of PH above the third.

As mentioned by Serge, FPT has a role to play here, but (I think) mostly in the fact that some problems naturally have more than one "input size" associated with them.