Perpetual stew-like stock using the fridge: is it a good idea?
I like this idea, a  lot actually, it would be an interesting experiment, but you would have to keep it on the gas.

Ignoring the safety concern of accumulating pesticides, which you mention, this would be like any stew which a quick Google search tells me should last about 3 to 4 days in the fridge. At least you don't want to cook meat in it.
Bringing it to a boil over and over again might help but you're also adding a new risk of contamination every time. Definitely not a long term solution.

In terms of flavors, you're basically making a stew out of anything; it could turn out great or horrible depending on what you cook but over time it would surely develop some strange off-flavors.

If you want to reuse your cooking water, I would do so in the same dish; for example, in the sauce. Or maybe you could keep it for a soup you want to make in the next day few days.