What is the highest power of 6 that divides 80! exactly?

6 = $2\times 3$, and 3 is the scarcer prime. Power of 3 in 80! = $[80/3]+[80/9]+[80/27]+[80/81]=26+8+2+0=36$ (power of 2 is 78, far more). So $6^{36}$ is the highest power.