Strictly speaking, you don't even need a partition for /. Just mkfs /dev/sda instead of /dev/sda1. Linux doesn't need a "partition". It only needs an fs to live into.
So, the lesser number of required partitions to run linux is exactly zero.
Some individual apps might get confused though, but that's their problem, not linux's problem.
You don't even need a swap partition, nowadays the kernel bypasses the fs when working with swap, so there's no penalty derived from having the swap space inside a file living in, let's say, /myswapfile.img
There are lots of situations where having separate partitions help, though. For example, having a single /boot partition allows for fancy things like sharing your kernel amongst distros (in most cases
there's really no point in having a different kernel for each, they all are linux, and it eases a lot of work), or when you want encryption. You will also needs a separate boot partition probably when your boot loader doesn't support the fs you use for /, that happens with newer fs's usually.
Having a separate /home eases the rare cases when you have to reinstall or update your OS.
You can have as many separate partitions as you want, using different fs's. The question here is are you looking for a solution for a problem or are you looking for a problem for your solution?