Although most experts believe that $NP$ is not equal to $P$, for a long time I believed that of the two directions of attacking the $P$ vs $NP$ problem trying to prove that $P = NP$ is the more fun one as it can be resolved by finding a clever algorithm, while showing $P \neq NP$ requires some abstract reasoning to show that certain problems cannot possibly be solved by any polynomial time algorithm.
Today it dawned on me however that other people have already done the abstract reasoning showing some problems outside $P$ and hence one actually might be able to show that $P \neq NP$ (provided it's true) by constructing an algorithm. Just take one of the known 'non-$P$' problems and show that it is in $NP$ by cooking up a certificate for it as well as an polynomial time algorithm that verifies it.
Although I have no plans of wasting my time carrying out this scheme, I am curious if there are any known problems for which one might try it. That is: are there any problems known not to be in $P$ but not known to be outside $NP$?
Yes. $\:$ The EXPTIME-complete problems are such problems.
More generally, for every time-constructible function f in $\:$n^$(\omega(1))$ $\cap$ 2^(n^(O(1)))$\:$, $\:$ the problem "Does the Turing machine M halt within f(number_of_states_in_(M)) steps?"
is not in P but not known to be outside NP.