How far down do I need to go when setting a 16 foot tall 8"x8" post?

I'd set it 24" deep (in concrete). Anything deeper wouldn't be very helpful, IMO.
Denis

I would recommend min. 4 feet to 4'6"
The formula for telephone poles at one time was length divided by 10 plus two feet, which would be 3.6 feet, but for an 8x8 that sounds too shallow. Four feet with concrete I think would be idea. These are just my own opinions.l

I'd agree with smearcase especially if you live in an area where there is frot in the ground in the winter. What will this be used for? If you use concrete put a few shovels full of gravel in the hole drop the post and then fill with concrete.