To find the greatest prime factor of a number, it helps to factor it completely into prime factors. One way to do that is to look for common factors that can be pulled out using the distributive property. I notice that $3^7+6^6$ has a factor of $3$ in both terms, since $6=2\times 3$. So I can write $3^7+6^6=3(3^6+2^6\times 3^5)$. Now I have a smaller number to factor: $3^6+2^6\times 3^5$. I see that this number also has a factor of $3^5$ in both terms, since $3^6=3\times 3^5$. So I can write $3^6+2^6\times 3^5=3^5(3+2^6)$. Now I have an even smaller number to factor: $3+2^6$. This number is small enough that I can calculate it directly: $3+2^6=3+64=67$. I recognize that $67$ is a prime number, so I don't need to factor it any further. Therefore, the complete prime factorization of $3^7+6^6$ is $3\times 3^5\times 67$. The greatest prime factor among these is $67$, so that is my answer. 
What is the greatest prime factor of $3^7+6^6$?