Fresh Mozzarella with the rennet?
Maybe I should ask the question this way:
What kind of home-made cheese can I do that is A) Quick and B) Does not require rennet or any enzymes.
If you don't have the time to do it properly, why don't you just go to your local cheese monger? I presume the aversion to rennet is that of choice to not consumer animal product?
Close to 100% of cheeses require enzymes. Of those that don't, you can make a cottage cheese/paneer type thing, which is quick, easy and tasty:
(you can sub whole milk for the heavy cream
)you can also press it into a block to give it more density if you want.
and you can make fresh cultured cheese, but that's not quick--you have to let it ferment a bit. if you want to try it, see pg. 291 of anne mendelson's book Milk. too long to paraphrase! but there are probably similar recipes online.
ditto mascarpone--easy, no enzymes, but also takes time b/c it's cultured.
if animal products are in fact the issue, you can buy vegetable rennet on line. if you're into making cheese, try using rennet! it's not hard--there are plenty of non-aged cheeses like feta or queso fresco that you can easily make with rennet.