Is every problem in NP?
There are plenty of problems where there is likely no way to verify a solution in polynomial time. 

Amazingly, the problem "is p a prime number" is in NP due to some deep mathematical theorem that shows that for every prime number, there is a polynomial time proof that it is a prime number. (For many, many NP complete problems the part that there is a polynomial time proof is quite trivial). 

A simple problem: Given an integer n, how many prime numbers p &lt;= n are there? There are ways to calculate this in $O(n^{2/3})$ and possibly a bit faster (which is exponential in the problem size). But there seems to be no possible proof that is less time consuming then finding the solution from scratch. so no proof that doesn't take exponential time. 

Here's a simple problem that I believe has an absurdly high time complexity: On the UK TV show "Countdown", the contestants are given six slightly random numbers, and another larger number, and are asked to form the larger number by combining any or all of the six random numbers using + - * and /. That's the simplest problem. The next, harder problem: Given six integers and some larger n, can all integers 1 ≤ i ≤ n be formed that way? The next, harder problem: Given n, are there six numbers a1 to a6 so that all integers ≤ n can be formed by combing a1 to a6?

And the final problem (as a decision problem): Given an integer k ≥ 1, and a larger integer n ≥ 1, are there k integers $a_1$ to $a_k$ such that all integers 1 ≤ k ≤ n can be formed by combining any or all of $a_1$ to $a_k$ using +, -, *, / ?