Basically the only parts the gun will continue to function without are the grip safety and the thumb safety. Oh, and the grips/grip bushings, and the front and rear sights.
Everything else, the gun needs. If any of them break or are missing, the gun will cease to function, or malfunction to the point of it becoming unusable. Basically it'll be guaranteed to go bang once, but it may or may not hit what you're aiming at, and the bang may be more an explosion than a gunshot
Plunger tube/spring/detents
Extractor
Ejector and pin (glue works, but only for a while..)
Barrel Bushing
Mag catch/lock/spring
Trigger
Hammer, hammer strut, and pin
Sear
Disconnector
Pins for trigger, hammer, and sear
Slide stop
Barrel, link, and pin (though you could make the argument that a properly fit barrel doesn't strictly need the link/pin...I wouldn't want to run it long without one though)
Firing pin stop
Firing pin
Sear spring
Recoil spring
Main spring
The breakage, failure or absence of any of these will cause problems. You can get around the mag catch by hand-feeding it individual rounds, but that's about it.
Some things don't fail as often as others, but it can't hurt to have pre-fit spares of each, especially if it's your duty/carry gun, or a gun you will otherwise rely upon for survival...
Odds are you'll never need that spare sear/disconnector/hammer, or a spare slide stop. But...better to have it and never need it, then to need it and not have it...