Let $X$ be the underlying space of an irreducible scheme. In particular, $X$ is non-empty.

Note that the closure of any point is a closed irreducible subset. We say that a point has codimension $n$ if its closure has finite codimension equal to $n$. So, the unique point of codimension 0 is the generic point.

Given a non-negative integer $n$, is it possible that there is a closed point in the closure of any point of codimension$\leq n$ but there is a point of codimension $n+1$ that has no closed points in its closure?

Note that if there is such an $X$, then $X$ is neither quasi-compact nor of finite Krull dimension (because then there is a closed point in the closure of any point).

P.S. a sub-question asked whether the fact that every point has a closed point in its closure implies the quasi-compactness of $X$. The answer is "no", as shown in the answer by A. Mathers.