I always thought demand was limited by [..] and most importantly income [..] I'm actually kind of surprised to hear that unlimited demand was a fundamental tenet of economics.

The problem seems to be confusion as to what economists mean by demand and "unlimited demand". What they mean by demand is not a static number but a demand curve relating how much of something you would buy depending on its price, i.e., not just what you're buying now but also what else you would buy if you could. Thus by "unlimited demand" they mean that there is always more you would buy if you could afford it.