One of the posts for my fence rotted away underground to where the post above ground could be simply pulled out. I fixed a ring with some lag bolts to the top of the remaining post (which is about 3 inches below ground level) and used a 7 ft pry bar to try to rip it out of the ground, but it turns out that the cement for the post was connected to the foundation of the house, so it's not going anywhere.
Beneath the initial rot, the post is still strong, so I can't break it up and pull the pieces out. I can't get tools like my drill deep enough to cut into it either. Maybe there's some tool purpose-built to drill/cut into the post, but I'm sure it'd be more expensive than I'd like to take care of a single fence post. The only other option I can think of is burning the remaining underground post away, or at least enough to compromise its structural integrity enough that I can pull it out.
Is there any concern / risk with doing this? The surrounding area is just dirt / concrete -- no grass or plants. The ground is extremely dry, though the fence post has naturally absorbed a bit of water in order to rot like this. The exterior wall above the hole is stucco. There's nothing flammable except for the fence post. I presume I should be using wood/similar based fire starters rather than gasoline given the plethora of gasoline-fire-starting mishaps documented well across the internet :)