Is it taking up a lot of space &/or sprouting? If not, you could always use it as a sturdy base for something? Maybe a place to sit or put a board on top as a place to harden off seedlings or grow stuff in grow bags?

JCB would do the trick, if you can get one in.

C4 would also do the trick ☺

You can get rid of the stump by building a small fire on the top of it if its wide enough. The roots will rot down eventually as long as it wasn't a sycamore. Some people drill holes in the top of the stump and pour diesel down them. Perhaps not a good plan on an allotment though...

If you're going to just kill the stump to stop it sprouting, don't use diesel,its really bad for the local environment. So is burning really, all that Co2 being released - use SBK. You'll need a drill with a medium sized woodbit to make wells in it about quarter to half inch deep, about an inch in from the outside edge, about 3 inches apart all the way round (assuming its a broad stump). Fill the wells up with neat SBK carefully (wear gloves, don't spill it anywhere else, its like water) cover with a sheet of plastic anchored down with bricks or something, or put a slab over the top to keep the holes covered. Job done... it won't rot down any quicker, but it won't sprout new growth either. And it won't poison your soil or you, so long as you don't spill any or drink the stuff.

Alternatively, if you have some friends with strong backs, it is possible to lever out a stump using a scaffold pole and levering all the way round until it shifts - I've seen it done, but it isn't easy...and okay on a stump say 6/8 inches across, but if its a couple of feet across, no go.

We are busy playing this game! We cut down a very invasive old laurel and are left with six trunks about six inches across all coming from the one huge root stock. We are cutting bits off as we go as we would really like it out, though we have bought Roundup Super Concentrate Stump Killer which works on the same principle as the SBK. At £16.30 it's not cheap but says it will kill up to 16 stumps. This is a last resort if we can't get the majority out, just how long the back-breaking work can hold out. Laurels shoot on anything so it has already sent out new shoots since we have been trying to get rid!
Dig a bit, find a root, saw it through, dig a bit further, the hole is already 3 feet deep! Best of luck, depends on what it was I suppose.

The reason I use SBK over the Roundup version is the active ingredient - its concentrated glyphosate in roundup, whereas its triclopyr in SBK. On current, up to date information, glyphosate is best avoided if possible because of its habit of binding to soil particles as much as anything - jury's still out re its carcinogenic effects, though there's a big court case going on in the States about it as a cause of cancer at the moment. But there is evidence it hangs around in soil for quite a while - found in Swedish forests two years after use, for instance.

Bamboo I hate diesel too but the questioner specified no chemicals, though come to think of it diesel is a chemical.
When we came here we had to remove an immature horse chestnut and a big conifer of some sort. We left the roots alone and they gradually rotted away. There have been no problems and no mushrooms. Its only been sycamores that won't lie down and die quietly...
Is SBK harmless then?

Nothing is harmless, Steragram, as you know - its only the level of harm we can keep down, or balance. And I don't think deisel is more harmless than specially formulated chemicals to do these jobs... even washing up liquid sprays aren't harmless.

What worries me about glyphosate based treatments is the fact we're all already full of the damn stuff, we're eating it all the time unless we only eat organic cereals, breads and so on. And glyphosate has been found in carrots grown on a plot 18 months after application... that doesn't always happen, it depends to some extent on the soil composition. I prefer SBK because its targeted, its only in those little holes in the tree stump, and nowhere else, But I wouldn't use it as a spray, despite what the usage instructions say ... only ever as a treatment applied direct to woody roots or stumps, and then covered.

Garden_gloves simply wants to remove an old tree stump because it's taking up valuable space on the allotment. I've used and axe and a pair of loppers to detach the main roots from the stump. Then I just went in there with brute force like King Kong and ripped out the stump with my bare hands. If it's too big, you'll have to call in a professional arborist to have the stump grounded out. I've used this service before and it's not unreasonable. You may get to keep the huge pile of wood chips that remain. Just remove the main stump & don't worry about the roots. They disappear naturally - not so for all those chemicals.