Just hot off the press: Have a pick-head type tool attached to a vacuum. The pick breaks apart the adjacent dirt around the stump. THe vacuum sucks the dirt to another spot on the property. Do this for a while. Bingo, tree caught with it’s pants down.
Cool the stump to absolute zero; when the tree atoms turn into Bose-Einstein condensate, seed this with some blancmange atoms and allow mixture to warm up; this should result in the tree root being transmuted to blancmange, which can then be scooped out using a plastic spoon. Blancmange and tree are right next to each other on the periodic table, so it ought to work.
Excellent, what do I win? Something large and giftwrapped? Sitting in a hole in the ground? All I have to do is dig it out and take it home? Sure, I’ll be right over 
Speaking from personal experience, in about five years you’re going to have a big soft spot in your lawn where the wood is rotting away underground.
Thankfully, Papa Bear has already solved this problem for you, and you may learn from his experience. Fun for the whole family!
Minature black hole in a magnetic container. Pass over trunk. Trunk becomes a singularity.
Antimatter would work too. But the ATF and the AEC would probably frown on it.
I’ve heard of belly button lint, but this takes the cake!
How about this novel idea:
Set a giant bell jar over the stump, pump it down to within a few bars of absolute, and suck all the juice out of the roots. That should wither everything holding the stump into the ground, and it’ll pull out like a weed in sand.
Second novel idea:
Freeze the stump solid with several applications of liquid nitrogen, then shatter it with a sledgehammer.
We have been wondering how we can get some liquid nitrogen for some time, just haven’t looked into it. Is it feasible?
Would that bell jar really work? How do you create the seal? Has this ever actually been done to your knowledge?
Feh. You’re just going to get a very very dry stump. Ultra dry wood may not be as resilient as normal wood, but this is not going to solve the problem.
I agree with Fuel, Pushkin’s winning, as far as I’m concerned.
Sealing a bell jar over the top to vacuum out the moisture might take longer than you think, since you’re essentially attaching it to a system whose chief purpose is to collect water from out of the soil.
And I’m not at all sure that liquid nitrogen would actually make wood all that brittle and I’m certain that you would not be able to use it to freeze just the root system in situ.
Hey, I already gave you the most practical method: explosives.
If you’re worried about debris & noise, just stack few dozen sandbags on the stump before setting off the dynamite. >Thud< instead of BOOM! You’re in Florida? Sand shouldn’t be a problem…
One more thought on this; you could invent a time-osity box to reverse the local flow of time and, using this, turn the tree back into an acorn - which you can just crush between your fingers as you laugh, madly and uncontrollably, with eyes insanely rolling.
There’s always someone who has to ruin it by going too far, isn’t there?
Yeah. Who can crush an acorn between their fingers? That’s just crazy talk.
So not only do we have to find a means of removing this stump, we now have to find a method of rendering an acorn susceptible to the pressure created by crushing within a human fist. That’s a whole other thread.
Duh.
If you’re wearing your nuclear-powered robotic titanium exoskeleton, or if your body is infused with superstrength nanomachines, then crushing the acorn is easy; you could probably crush three or four acorns in the space of a morning.
Believe it or not, you were not the first to think of this. My brother pondered a way to reverse the growth of the tree just yesterday, on the 7th day of our brainstorm. Whew, I had to pull his mind back into reality on that one before I lost him forever…
By the way, our pro chainsaw could not get through this ficus yesterday, too irregularly shaped and thick.
Out of anger, We attached a 2-ton come-along between the remaining trunk and its roots to use its power against itself… but bent the tip of the hook…
We have initiated phase 2: Pipe Bomb.
After that is to soak it in deisel fuel and turn it into a slush puppy.
you know what was cool? Our power drill with a wood bit eats right through the heart of the tree with no problem, and twisting it in the hole produces a HUGE gouge in the tree… hmmmm???
I await the news of your Darwin Award with great anticipation.
Use the Force, Luke.
Or, give it to a 5-year-old to play with. They break their toys in 30 seconds flat.
Or, just piss on it, every single day, until it dissolves in your vile, foetid juices.