I'll offer some thoughts, having run loads of Hondas over loads of years.
Plus some questions.

There must be some force pushing the sprocket longitudinally along the gearbox shaft, away from the gearbox.
Normally in my experience there's very little sideways force, which is why the retaining washer isn't particulary substantial.
As you've no doubt experienced they last a long time, tens of 1000s of miles, and maybe become a little thinner on the internal lugs.
Also, I'd say the sideways force must have some 'hammer' effect to destroy your latest washer so quickly.

So what could be doing that? My first thoughts are: rear wheel bearings, or swing arm bearings.
Have you noticed any change in handling at all?
Maybe even a swingarm that's broken somewhere. (Sounds like you've done a lot of off-road. Particularly fast or heavy-going?)
With rear wheel off the ground, apply a strong arm to the tyre and see if you can make any side-to-side movement.

If handling is definitely OK, does the chain have any seized links? (Although not very likely to cause this). Check with rear wheel off the ground (see below).

Rear sprocket carrier bearing?
Don't know the cush-hub arrangement on the TA, if it has one, but is it OK?

Is there any longitudinal free play on the gearbox shaft? (In-and-out). There should be none (or maybe very little).

Put everything together, but leave off the retaining washer. Adjust the chain to the correct up-and-down play.
Set the rear wheel off the ground with bike vertical.
Start engine and select 1st gear. Watch what happens. With no drive force being transmitted through the chain (just enough to rotate the wheel) the sprocket should stay in place. At least for quite a while.

If not, the way it behaves may give a clue.
Does the chain bounce a lot forcing the sprocket off?
Does the gearbox shaft move in-and-out noticeably?
Watch the chain in motion. Is there a single link badly damaged and out of line? Or a pair or three links together?

As another test, try winding a length of mild-steel wire around the shaft in the groove, enough to fill the groove and leave a turn or so raised above the level of the splines. Wind it quite tightly and twist the ends together firmly.
That should be enough to hold the sprocket on for quite a few miles. In Turkey we did that to a fellow-traveller's Honda after he lost the two retaining bolts and the washer. It got him to Iran OK, quite a few hundred miles.
You could try solid copper wire if no steel, should be just as good.
So, if you go only a few miles before the whole lot comes off, there's definitely something wrong with the whole alignment of the two sprockets (a frame/swingarm/bearing thing), or the bearings of the gearbox shaft.

That's about all I can think of for now.
Hopefully others will have more to offer.

Best of luck!

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TTR250 - London to Cape Town