We were clearing vegetation away from the side of the house and found a tree growing right up against the foundation. I had lopped off the top of the tree when we moved in, but it sprouted around the cuts and continued to grow. Unfortunately this tree had to go, so we used the drill and fill method to stop its growth and allow the stump to rot away.
The Method
Cut the tree a few inches above ground level, as flat as possible. You might need to cut the tree down and then trim the stump flat. Drill 1/2-in to 3/4-in holes vertically in the stump. For every two inches in diameter, drill one hole. Distribute the holes evenly around the stump, drilling down about 4–6 in. Fill these holes with kosher salt (epsom salts work fine, too). This dries out the stump by drawing away moisture. Cover the stump so rain doesn’t dilute the salt. You may need to refill the holes periodically. Over time, the stump will die and rot away.
It isn’t fast, but it will work without nasty chemicals or a huge piece of gas-guzzling equipment.