We can live without a lot of parts of our body. We don’t *have* to have legs or arms, although life is much easier if we do have them! Some of our organs are necessary, like heart and brain, and at least some lungs – but as any lung cancer survivor will tell you, you can survive even with a lung removed! We have two kidneys, so losing one is possible and you’ll still have a spare. I’m pretty sure we can live without our appendix or gall bladder too (my mum doesn’t have either of those anymore). Same for reproductive organs – I don’t think those are necessary for life, either! We can live without our spleen too… but gosh, I’m beginning to get depressed now. There isn’t much of this poor person left!
Loads! It mainly depends on what sort of life you’re willing to live… Yo mentioned lots of the organs we can carry on without, but we can use machines to replace a few others too. Dialysis machines will do your kidneys’ job, for example, and there are combined heart and lung machines that will breathe for you and keep your heart beating. We don’t have a good machine to replace it yet but your liver will naturally grow back even if three quarters of it is removed, and you might even cope with surprisingly little of your brain – some people manage to live incredibly normal lives with fluid filling bits of their skulls where most of their brain should be (this guy, for example http://www.thelancet.com/journals/lancet/article/PIIS0140-6736(07)61127-1/fulltext), though that’s extremely rare. Again, though, it comes down to whether all this really counts as living!
There are lots of parts of the body we can live without (but personally, I would rather live with all of them). Personally, I’m always surprised at how much damage the human body can take and still go on!