An example of a computable problem that is not in P I am trying to find a simple example of a problem that is computable but not in P, I know very well that it would be enough to get one in NEXTIME-complete however the problems that I find in this set are very artificial
As Yuval Filmus already mentioned in the comments, because of the time hierarchy theorem, we know $\mathsf{P} \neq \mathsf{EXP}$. Hence, any $\mathsf{EXP}$-complete problem is an example. These include, for instance, generalized chess and checkers.