<A HREF="https://www.math.ucla.edu/~ynm/papers/1602-002-1.pdf">Kleene’s amazing second recursion theorem</A> by Moschovakis contains:
> A list of some (18) of the most significant applications of the
> <A HREF="https://en.wikipedia.org/wiki/Kleene%27s_recursion_theorem#Kleene's_second_recursion_theorem">Kleene's Second Recursion Theorem,</A> in a kind of “retrospective exhibition” of
> the work that it has done since 1938. [...] These were selected
> because of their foundational significance and the variety of ways in
> which they witness how the recursion theorem is used.

A <A HREF="https://www.math.ucla.edu/~ynm/lectures/2009csl.pdf">talk</A> by Moschovakis gives an overview.