I am aware of groupoid categories. Recently, I came across the category of all the groupoids in JP May's A Concise Course in Algebraic Topology, wherein he describes:
Taking morphisms to be functors, we obtain the category of groupoids.
However, I am a bit confused: Don't we require that for any two groupoids, say $G$ and $H$, the functors $G\to H$ form a set? How to ensure this?
Groupoids appearing in algebraic topology and algebraic geometry quite often refer to small groupoids, and many people make this tacit assumption. In fact, groups are assumed to be small 99,9999% of the time, and groupoids have been classically defined as groups in which the operations are only defined partially (the category notion and interpretation came later).
But in any case, you can also consider the category of all groupoids, or in fact all categories, when you use the correct set theory such as Grothendieck-Tarski set theory. Usually we fix two universes $U \in V$. Everything in $U$ is considered to be small, everything in $V$ is considered to be large, but of course both $V$ and all of its elements are still just sets.
The classical statement "The category of small categories is well-defined" can be then generalized, or "upsized" to: The category of $U$-small categories is a $V$-small category. But you can now repeat the game and choose yet another universe $V \in W$. Again, the category of $V$-small categories is a $W$-small category. etc. There is no limit.
Now, not every category is small, but it will be $U$-small for some chosen universe $U$.