I am trying to find a way to boil 1-2 quarts of water for 10 minutes and want to be absolutely sure that the pot doesn't leak any chemicals into the water. I have tried 3 stainless steel sauce pots but they all rusted (and one rusted after a single use). What kind of sauce pot can I get that is guaranteed not to rust or break due to temperature and not leak any chemicals into the water? Price is less important than finding the right item (I'm only going to buy 1).
Just use a glass pot, Corningware make some decent ones that can be used on a stove. But any glass pot designed to work on a stove with direct heat/flame exposure should work in this case.
If you are worried about uneven heating or hot spots on a gas range due to less conductive glass, you can use a cast iron or aluminum heat diffuser plate under the glass to even out the heat.
Quality austenitic stainless steel* should certainly not rust from boiling plain water (excepting MAYBE some rust in places like handle weld spots - the metallurgy is upset in these spots, and usually they don't touch the food anyway.), given that cookware is made from it that is perfectly dishwasher proof, and won't rust if salted water is boiled in it. If such items quickly rust, assume there are aggressive chemicals (eg from a cleaning product you use) or electricity (eg insulation being marginal on a stove, and stirring cook+metal/wood utensil providing a ground path) involved. In some odd cases, keeping the cookware in water with items made of very dissimilar metal (eg bare copper) could accelerate corrosion due to the whole setup acting as a battery. Another possibility is getting rusty water from cleaning non-stainless things onto stainless surfaces - think of stainless steel more as a material that usually doesn't start rusting but can rust once it started.
Alternatively, consider ceramic coated nonstick pots.
*There are a few basic kinds of stainless. The other major type (in kitchen applications), martensitic, is used for things that need to either keep an edge or resist permanently bending (knives, graters, springs...), and is far more prone to corrode. However, there would be zero reason to make a saucepan out of it.
Aluminum won't rust. I threw away the ceramic (Corning) pan I had as it was slow to heat and would burn anything but water. A non-magnetic stainless steel pan ( austenitic / 18-8 ) will not rust when used in a kitchen for food. If used to boil hydrochloric acid or boiling salt solutions down to a mush-like consistency , you can get rust . One can get rust spots at a spot type weld; this is caused by sensitization near the fusion line ,especially when the stainless has high carbon and low chromium . I have never seen it in cookware but substandard stuff from China may have this problem.