What is the easiest way to remove a large root in a post hole I'm digging?
A reciprocating saw works nicely, with the proper blade. You can buy a tree cutting blade for work like this, 10-12 inches long, with long sharp raking teeth.

We had an ice storm, and the clean up of hundreds of large branches and trunks was done entirely by reciprocating saw, wearing out only three of those blades in my Porter Cable. Some trunks were over a foot in diameter. (My chain saw chose that time to be singularly uncooperative.)