Carry your wallet and cell phone and you're golden.
I bike commute 8 miles each direction and don't carry a tire repair kit or pump. I've never had a flat in my ~15 years of bike commuting, probably because I ride "puncture-resistant" tires (like the aforementioned gatorskins, but mine are
panaracer ribmos). They are a little heavier than the tires on my race bike, but they never get flats. My race tires get a flat at least once per season, and I don't even ride that bike very often.
I do carry a flip-out
multitool that lets me adjust almost everything on the bike on the fly. Occasionally I need it to adjust my fenders or brakes, for example. You have to know what your bike needs in order to get one that has all of the right tools, but most people really only need a screwdriver and the correct three sizes of allen wrenches.
For catastrophic mechanical failures, I can either carry the bike home or call for help. There are buses that run along most of my route, so even if I couldn't call my wife to pick me up I could wait 30 minutes for the next bus. I've never had to do this, even when I've crashed on my commute and had to spend 15 minutes fixing the bike well enough to finish my ride.
I also recommend
front and rear lights if you ride in anything other than full daylight. There are lots of decent sets on amazon for under $100, but I'd avoid the very cheapest ones.
My big U-lock lives downtown at work. It's too heavy to bike around with daily unless I know I'm going to stop on the way home and need it. I have a lightweight cable lock that I use places like the grocery store and library, but I live in a low-crime area and am not really worried about bike theft.