What is the largest set for which its set of self bijections is countable? I recently came across a problem which required some knowledge about the self bijections of $\mathbb{N}$, and after looking up how to construct some different bijections I came across the result that the set of self bijections of $\mathbb{N}$ is uncountable. 
And this got me wondering, what is the largest set for which its set of self bijections is countable? This obviously holds true for any finite set, but what is the last example of a set whose set of self bijections is countable?

There is no such maximal set, because $\aleph_0=|\mathbb{N}|$ is the smallest infinite cardinal.  
