Cut off new growth &/or skin the bark off the tree trunk(10 high is a tree).
2) Drill 1/2 inch hole straight down in good wood of the tree, about 3 inches deep. Pack with table salt the bottom 2 inches, then add melted wax to seal the salt in.
3)Use Toxic poison.

Cutting off any root suckers and new growth is really the only non-toxic method if you don't want to grub the stump and roots out by hand. It is certainly not an instant fix - you will have to repeat doing this for sometime, but if the tree cannot photosynthesize (and it needs leaves to do so) it will evetually starve to death and die, even all the roots.

There are other methods of both killing off trees and any root suckers and removing the stumps but none are considered completely organic as they involve the use of chemicals or petroleum products.

Not all types of maples will regenerate from the roots if cut down....you may not have to do anything :-)

My 'mad-scientist' method used to 'melt' several stumps.

1] pour one or more bags of steer manure over top of stump.
2] cover with heavy black plastic
3] cover with soil and plant vines .... [I used honeysuckle]

In about 2 years, when I re-did that part of the yard, the stumps were gone. My stumps were not as large as yours, and were a softer wood, so that probably made this method work so well.

If you keep cutting off new growth the stump will eventually die. I don't see why you think this is a big deal, two minutes every couple of weeks with a machete or ax will accomplish this.
If you also want to remove the stump you can keep working on it with a mattock, lots of good exercise.
Or you can drill holes in it and pour potassium nitrate in the holes, which will make it quickly rot out, or if you live in an area where fires are permitted you can burn it out, the potassium nitrate is an oxidizer.

I'm not so sure that the potassium nitrate works as well as advertised as a stump rotting agent, having several stumps 5 years later with no visible difference. However, the stuff is absorbed into the wood, and if you drill it in, wait 6 months, then light a fire and use a leaf blower, you can burn the thing out in an hour or so. Lots of fun, sparks flying all around.

Some Maples sometimes do produce suckers after being cut down while others do not do that. The simplest, easiest means of control, if suckers do appear, is to cut them off.
Potassium Nitrate would not be an acceptable organic means of control and burning the stump would be prohibited in many places because of the large amounts of air pollution produced.
Laying animal manure over the stump, or blood meal, would be acceptable organic methods of getting rid of that stump, if that is desired.

Do what we did...get a couple neighbors and haggle with a tree service guy to come grind it. Those guys are pretty notorious for going door to door when they are in the neighborhood, and I hardly blame them. Driving all over town to customers houses can eat up half the day (so much easier to sorta plant yourself in a neighborhood and knock out 3 or 4 jobs).

Thanks for all of your input. I actually dont care if I get rid of it, I just want the roots to die that are choking out a couple of my beds. So far it hasnt tried to grow back, but it has only been 12 days. I will just keep it clipped if it does, at least for this year.