First note to the following well known theorems:
Theorem (1): The notion of "$x$ is a strongly inaccessible cardinal" is first order expressible and $\Pi_{1}$.
**Theorem (2):** The notion of "$x$ is a measurable cardinal" is first order expressible but not $\Pi_{1}$.
Theorem (3): The notion of "$x$ is a Reinhardt cardinal" is not first order expressible. Now there are some questions here:
Question (1): Are larger large cardinals more complicated in first order expressibilty? Is there any exception?
Question (2): Is there a non first order expressible large cardinal weaker than Reindhardt cardinal?
Question (3): What is the largest $\Pi_{1}$ - expressible large cardinal? For example the notions of being a Mahlo or weakly compact cardinal are first order expressible and $\Pi_{1}$.