just take a small cotton towel and if need be just keep wringing it out, friend of mine when made homeless managed to get the use of a caravan and he learnt a few tricks there, he made do with a hand flannel to quite good effect (apparently!)
i came accross a small jobby in a surplus shop for 50p and thought i'd give it a try, worked well enough but scrubbibg your back makes it look like your climbing out of a straight jacket
I use what is basically one of those outrageously expensive, miracle fiber backpacking towels, but found some at a dollar store for... a dollar. They don't come in fancy colors or have a grommet in the corner for hanging, but so what? I can do that myself if needed. These babies suck up water like nobody's business and dry fast. I should add that I got the idea of looking for them in the dollar stores from Hoodoo.
Forgot to say that if you can't find the el cheapos like Hoodoo and I, try the auto stores. Around here, you can buy an artificial chamois cheaper than the fancy BP towels (same thing, BTW).
If you're in the UK, get to "Poundstretcher" They have miracle fibre cloths for about £1.99.
OK they're bright orange and they haven't got "Berghaus" written on them, but for around £9 cheaper , I've a couple. One in my washbag and another in the cook kit. If you need a bigger towel just sew a couple together...Job Done
Another idea is just a square of pertex, which packs down to nothing. Its nature is to abosrb water and spread it across the whole fabric. Use it to blot rather than rub and then let it blow in the breeze and it dries very quickly. I would not want to use it after a shower but for a "top & tail" freshen up wash in field conditions, it's fine. I got mine from Penrith Survival about 20 years ago and it's still going strong.
I have known people use a pertex windshirt (e.g. unlined Buffalo) for the same purpose.