That is, how much of my innards could I be missing and still live a (relatively) normal life? Assuming I take good care of myself (Willy-nilly removal of organs aside) and take proper precautions, what can I do without? The first things that come to mind are organs I’ve got two of: kidneys, lungs, etc. Once those are gone, could I live with, say, half of a liver and spleen?
I suppose the question comes down to how resilient and/or redundant the human body is.
Gall bladder
Appendix
IIRC, the liver can regenerate itself so I suppose you could get by with half of one.
If ya got the faith, you can live without any of your body.
and if you don’t then you won’t want to! 
(they say it gets really hot.)
hell, what do you need, lessee. heart, one lung, one kidney, pancreas, liver, intestines (and maybe only small), brain, skin.
what can you do without? spleen, gall bladder, stomach (saw a story once bout a man who swallowed drano in a botched (verrrry botched) suicide attempt- they took out his esophagus and stomach, and his small intestine was threaded up over his rib cage (under the skin, of course) and up his neck- food went right into his duodenum, and he had to massage it down his throat and ribs, like he was stroking a very long, very imaginary beard- freaky), eyes, lower jaw, ears, legs, one arm (maybe both, if you have a helpful manservant- but how would you ring the bell to summon him?), probably large intestine (as long as you make sure to stay hydrated), one’s neener bits, not sure about which glands you need (not even sure which glands are organs, to tell the truth), most of your peritoneum, definitely all of your perineum, most of the muscles (all of the voluntary ones, i think- again, assuming you have a faithful manservant), your spine (providing you have a way to support your spinal cord- perfect job for a manservant), and your cranium.
provided you have a helmet.
oh, and by the way, I Am Not A Doctor (but i would get rid of all that spare baggage for ya, for a small fee).
and, if you ever need a trusty manservant, i’m your man. small fee. trust me- you wouldn’t have to ever even ring the bell. i’d always be at your side, valient and dependable.
jb
You can also get rid of half of your brain like this girl and still do just fine too http://abcnews.go.com/sections/GMA/GoodMorningAmerica/GMA020712Encephalitis_girl.html
This is going to be quite the weight loss plan.
You’re going to need some bone marrow, to keep your blood supply going - will your skull bones suffice for this, or should you consider keeping, say, one limb?
In keeping a ‘semi’ normal life I would think you would need your limbs - perhaps a few toes can be hacked.
You don’t need your tonsils, either.
Shagnasty,
What did they put in the area that had formerly contained brain?
As for Catalyst, there’s a Stephen King story about a doc who eats most of himself, Limbs and stuff. I think he was waiting for a no’ 10 near King’s Cross.
Well, technicly, you could get away with just being a brain…they’ve kept monkey and dog brains alive on life support systems for a couple of days. To live longer than a few hours, though, you’d need a “host” creature to provide you with nutrient-filled blood.
Of course, being a disembodied brain on a life support system with no kind of sensory input whatsoever goes a long way from living a “normal life,” as you put it, but I thought you’d be interested anyway. 
Hee’s something else along those lines that you might find interesting; The Injuries of Darth Vader.
Ranchoth
Cecil comes through again. If surgeons remove part of your brain, how do they fill up the space?
I think you can do without hair and ears as well. Planning on becoming a cyborg are we?
Not just any cyborg; I’m gonna be ROBOCOP! Except I’ll walk faster (I’d be built with better special effects) and I’ll get satellite reception for my visor. And maybe a built-in Mr. Coffee for the rest of the department.
Don’t forget the highspeed wireless Internet connection and data gloves. 
from Ranchoth’s link-
“Respirator is required, with a full-time attendant for the patient.”
seriously. i need a fucking job. i would be a good manservant. hell, Catalyst, i would gladly make coffee for y’all. i’d even let you call me Jeeves or some shit. what do i have to do?
jb
You can get rid of your tongue, but in order to eat you’d have to cut your food into veeeery small pieces, so you could swallow them instead of chewing.
Actually, you can probably do without a jaw and teeth.