I searched for primes of the form $p^p+2$, where $p$ is prime for a range of $p \le 10^5$ on PARI/GP and found that 29 is the only prime of this form in this range.
$(1)$ Is $29$ the only prime of the form $p^p+2$, where $p$ is prime?
$(2)$ If not, then are there a finite number of primes of the form $p^p+2$? Can you prove/disprove this?
Edit: Since $p^p$ grows really fast and primes get rarer and are spread farther out for large numbers,
I conjecture that $29$ is the only prime of the form $p^p+2$ where $p$ is a prime.
With pfgw, I checked $$p^p+2$$ for the primes from $\ 3\ $ to $\ 24\ 001\ $
The only prime occured for $\ p=3\ $ Hence if another prime of this form exist, it must have more than $\ 100\ 000\ $ digits