Can the expression $$n^n+\varphi(n)$$ be a prime number for some integer $n>19$ ?
For $n=1,2,3,19$ and no other positive integer $n\le 3\ 000$, the expression is prime. A further prime of the desired form must have more than $10\ 000$ digits. For $n>2$, only odd $n$ need to be considered because $\varphi(n)$ is even for $n>2$
Moreover, I search a prime factor of the composite $283$-digit number $$f(133)=133^{133}+108$$ Probably , there is no prime factor with $20$ digits or less.
Since you found no prime with composite $n$ for $n \leq 10^4$ the only explanation that I am aware of is that $\phi(n)$ is very likely to have form $\phi(n)=2r(n)$, where $r(n)$ is either a prime dividing $n$ or is divisible by a prime that also divides $n$.
So, further study of function $\phi$ is needed, that is, to determine for which $n$ we have that there is a prime $p$ that divides both $n$ and $\phi(n)$.
It could be that quite often we have $\phi (n)=m(n) \cdot p(n)$, where $m(n)$ is even and $p(n)$ is some prime dividing $n$, because then we could not have $\phi(n)=m(n) \cdot p(n) | (n-1)$ because of $p(n)|n$ and that would be in favour of Lehmer´s conjecture.