I know you can get by on one kidney, a part of one lung, and without a spleen, gall bladder and appendix. What else can be taken out and still have the person live normally, with no artificial support?
The liver is the only internal organ that can completely regenerate itself. It is possible to remove over half of it and it will grow right back.
I lung, spleen,1 kidney,appendix, gall bladder, thyroid, thymus, ovaries, these can all be removed.
If you are asking how much of the internal parts of the body is not needed to survive I’d add eyes, most of the stomach, good chunk of the intestines, teeth, mammary glands, ovaries, testicles, the rest of the reproductive system. I don’t know how many of these you’d consider organs
You can live without a colon. The small intestine takes over most of its functions. Most of the stomach can also be removed.
You can survive without reproductive organs. Sex hormones are produced to a limited degree by other organs.
As for other organs, it depends on what you you mean by a “normal life.” You can survive without a rectum or bladder but that requires management of colostomy or urostomy bags.
What kind of quality of life are you looking for here? And what constitutes “artificial support”?
You can live with a single lung, and surprisingly it doesn’t impair function or fitness much. So I’d amend that to the list unconditionally. Not just “part of a lung,” you can lose the whole lung and still live a normal life.
Uterus, ovaries, and fallopian tubes, as well as testes and prostate are optional organs, although there may be hormonal consequences. But enough people do it that I think it falls under “normal life.”
You can live without a large portion of your liver, but it will likely grow back, so that’s kind of cheating.
You can live without your large intestine and most of your small intestine without necessarily needing artificial openings into the body (ostomies). If they only have to take out part, they can often attach the two cut ends to each other and you’ll still excrete normally. If they have to take out the whole large intestine, including the rectum, you can get a procedure done to create a pouch in your small intestine that connects to your anus, and that eliminates the need for an ileostomy.
You can live without your fat (which is technically a tissue, not an organ, but I thinkthere’s an argument to be made that “fat” in aggregate works like an organ. The only thing that keeps us from calling it an organ is that we’ve defined organ as a thing having more than one type of tissue in it.)
You can live without a bladder, but you’ll need another way of eliminating urine - usually a urostomy, but sometimes they will make a neobladder out of a section of small intestine instead, removing the need for an ostomy.
You can live without a pancreas, but you will need to take both insulin and oral pancreatic enzymes several times a day.
You can live without a stomach, but you’ll need to get your nutrients through an IV or through a tube that goes into your small intestine through your abdominal wall.
“Mrs. Brown, can Billy come out to play baseball?”
“But you kids know he has no arms or legs.”
“That’s OK, he can play second base.”
I think a lot of people can live without their brain. Seems they never use it. 
Some people can live normal lives with just half a brain, after a hemispherectomy for example.
Here’s one idea for weight loss that will NEVER be popular.
Short answer: Quite a lot.
A number of people have lost their tonsils when they were children.
This thread reminds me of the book Never Let Me Go, by Kazuo Ishiguro, which is about clones who are kept to be living organ donors. IIRC, a clone is said to be “completed” (i.e., dead) after four donations, and I think there’s a reference to one clone who was “completed” after only one or two donations, presumably because one of them was a heart.
If the OP wasn’t from over a year ago, I’d put in a grinning “Need answer fast?”.
Better late than never!
You could survive with no organs at all if you provide vascular inflow (via carotid arteries) and outflow (via jugular veins) of blood to your decapitated head, maintaining proper pressure and balance of oxygen, electrolytes, glucose, hormones …
This was demonstrated rather grotesquely in the 1970’s when Dr. Robert White transplanted the head of one monkey onto the body of another monkey. It lived and remained conscious, able to see, hear, taste, bite, show emotion, etc. Granted, that was one seriously pissed off monkey who could no longer fling feces and whatnot … so, not exactly “leading a normal life”, but it survived.
Technically, the monkey head used vital blood constituents manufactured by organs from the other monkey’s body, but there’s no reason those constituents couldn’t be manufactured and supplied artificially.
Who knows, in the future, if synthetic motor and sensory neurons can be 3D printed, perhaps an organ-less monkey head really could play the bongos and lead a normal life. Maybe your head could, too.
That monkey was ahead of his time.