I was thinking on this theoretical experiment for a while, but nothing useful came into my mind.
We have a "resistor" (rather a material like plastic) with a resistance expected to be very high, say 10^5 Mohms. Is it possible to measure the accurate vaule of the resistance with a max. 25V DC power supply (higher voltage damages the resistor) and a multimeter with max. 0.5μA resolution? (This is the main problem: if you connect the resistor and the meter serially to the scource, the meter won't show any current: \$ I = \frac{U}{R} = \frac{25}{10^{11}} = 2.5 \cdot 10^{-4}μA. \$ ) You can use any simple circuit element like resistors, capacitors, diodes etc., maybe really simple ICs like amplifiers at max. The solution does not have to be a circuit: the only aim is to get the accurate vaule of the resistance.
Best Answer
You can do what you want, but it won't be simple. Scratch that, it will be fairly simple, but it won't be easy. As von Clausewitz said, "In war, the important things are very simple, and the simple things are very difficult." It applies here as well.
Actually, it's not as bad as it could be. Your nominal current is 250 pA. You can get op amps with bias currents in the 1 femtoamp range, so monitoring your current is certainly possible. I recommend you start with Bob Pease's "What's All This Femtoamp Stuff, Anyhow?" http://electronicdesign.com/test-amp-measurement/whats-all-femtoampere-stuff-anyhow followed by "What's All This Teflon Stuff, Anyhow?" http://electronicdesign.com/test-amp-measurement/whats-all-teflon-stuff-anyhow
The devil (or God, depending on your background) is in the details, and at these current levels details matter a whole bunch. You'll need to clean your apparatus with alcohol to get any fingerprints off, and grounding will take on a whole new meaning for you.
But it's certainly possible.
Basically, you're talking about making a picoammeter, and you can buy those commercially. Here, http://www.rbdinstruments.com/products/picoammeter.html?_kk=picoammeter&_kt=3ce3e0b6-02d3-4edb-8960-2273c533a323 for instance, is a cute little USB unit, but I'd be surprised if you can get it for less than $1500 - or maybe more. You can try eBay, and get rather cheaper than that.