I run my AR's pretty hard, a few of which are full auto and sometimes in need of repair at the range. So I keep a small range box with me stocked with the parts I most frequently need, starting with tools that are essential: Allen wrenches, a gunsmith screwdriver set, some decent punches, a brass hammer and drift, a small socket set, a good flashlight, and basic cleaning and lubricating supplies. I keep a good AR stock wrench on hand at all times as well.
Now as far as AR parts go, the small parts in the lower receiver and BCG are the ones that most commonly break or get lost. So I keep extra hammer and trigger springs, a mag release spring, buffer retaining pins and springs, and detents and springs for a start. I also keep an extra firing pin or two on hand along with an extra bolt cam pin and several firing pin retaining pins (which I am notorious for dropping on the ground to never see again)
. If you have a good carrier that has a properly staked gas key, then there's no reason IMO to keep an entire BCG on hand as a spare. I do however think either a spare bolt is wise, or alternatively you can keep a bolt rebuild kit on hand that has gas rings, extractor springs, extractor pins, and a new extractor. I personally like to keep extra bolts around because they only cost about $50 and it's just faster to swap bolts quickly than rebuild your bolt if it goes down while you're shooting.
I also like to keep extra buffers on hand of different weights which can easily be swapped out if/when you encounter feeding problems of any kind. I've got enough AR's now that I pretty much have extra buffers of every weight on hand. If you're budget minded however, having an H3 and carbine buffer is all you really need since the weights can be swapped between the two and configured into H1 or H2 pretty quickly. Extra buffers probably aren't essential if you're just running a stock carbine buffer anyway and not running suppressed or full auto. If however you move to a heavier buffer I'd recommend at a minimum keeping an extra carbine buffer on hand in case you start encountering feeding problems.
So that's what I keep around to keep my guns running. It might sound like a lot, but it's really not at all. If you weren't keeping additional buffers around then the rest of the parts including an extra bolt can be acquired for less than $100, and if you have those parts there are few problems you'll encounter that you won't be able to repair.