Let $n$ be funny. Clearly $2\nmid n$. Check that $n=3^4$ is funny, but $n=3^5$ is not. If $p^2\mid n$, $p>3$ prime, then $p+2$ prime forces $p\equiv -1 \pmod{3}$ but then $p^2 + 2 \equiv 0 \pmod{3}$. Similarly, if $pq\mid n$, $p,q>3$ primes, then $p+2, q+2$ primes force $p,q\equiv -1 \pmod{3}$ but then $pq + 2 \equiv 0 \pmod{3}$.
Therefore we can have at most $n=3^4p$, with $p>3$ prime. However, if $p\neq 5$ then $5$ divides at least one of $p+2, 3p+2, 9p+2, 27p+2$, while for $p=5$ we have $81p+2 = 407 = 11\cdot 37$. So in fact we can have at most $n=3^3p$, and according with the above, only for $p=5$.
Indeed, $\boxed{n = 3^3\cdot 5 = 135}$ has $8$ divisors and is the only funny number with as many divisors.