As Iwire posted, it depends on several things. The concerns include soil, lateral load, vertical load, grade of construction, and environment. You might need a stronger pole, engineered pole, more depth, soil enhancement, added footers, and enhanced lateral support.
That said, some rules of thumb are: For good to very good soil and light to average pole loading, 10% plus 2 ft. For heavier loading and average to good soil, 10% plus 4 ft. Use at your own risk as use of these rules should be tempered with experience.
You should also be concerned with the pole class as more vertical load will require a bigger pole and you may even have to have a footer bigger than the pole. Other concerns are enhancing the soil to provide more stress withstand, both vertical and lateral.
I have designed installations where 3-4 ft diameter poles had heavy 5 and 6 ft diameter concrete footers and concrete backfill. You could also have 1/3 or more of the pole in the ground in some cases. Better to ask someone who does it for a living than guess on your own or use some rule of thumb you are not familiar with. Remember you not only have to consider the hole depth but the pole strength, loading, and support as well.
