Did this twice in Scottsdale a few years back.
#1. Used some large (more than 1") and long (18" +) old style auger bits to drill deep holes in a circle pattern (and within the circle) about the overall diameter of a big power auger like they use on Bobcats or the trucks that plant telephone poles. Did that to open/weaken the circular area as deep as possible. May need to do two or three circles, depending on diameter of stump. Brought in the power auger and drilled out a big hole(s) in the stump. Used chain saw to cut sideways from the holes - to get at little feeders and shred as many as possible. Pulled the remainder out with a cable/chain and the Bobcat.
#2. Dug deep and wide around the outside of stump with a rented medium sized backhoe - went as deep as feasible, cutting some roots in process. Used the chain saw to cut down and across the many feeders. Dug some more with backhoe. Cut some more - until through most all of the underside. Yanked out the remainder with cable/chain and backhoe.
A lot of work either way, but let your brain and the machines do the heavy stuff. It was in summer - good to have the pool nearby.
