shorts (two pair minimum: one to wear, one in the wash)
bottles and cages
seat pack with spare tube, patch kit, inflator, maybe a multi-tool
stiff-soled shoes and clipless pedals
bike-specific tools, if you want to do your own wrenching
rain gear and/or cold-weather clothing
IMO you need the first three items for even the shortest of rides. Add the bottles, cages, and seat pack by the time you work up to rides of an hour or more, and/or venture more than a few miles from home. From there on it's largely a matter of what kind of riding you do and personal preferences. Most people tend to build up their collection of jerseys, tools, and rain/cold gear over the years.
I wouldn't consider a helmet an accessory--it's a necessity. I've fallen many times in 30 years, and I think I've only hit my head once, but I still don't go down the driveway without one. Ditto a water bottle, at least for rides of more than about 20 minutes, and the tool kit with patches and pump (are there really people who ride WITHOUT that?) I average a flat about every 100 miles here in Big Thorn Country--I'd be walking every second or third day. Here in the desert, anyway, I'd put sunscreen on the necessity list as well. I've been riding without it all my life, and the last few years I've had to have several "probably sun-related lesions" removed from my arms and legs. Had a nice tan back there in the '80s, though.
Now that I think of it, a lot of the rest of the stuff mentioned here, like shorts and shoes and gloves, are pretty much necessities, too. First thing I'd put on the "nice, but not necessary" list is probably a computer. You don't need it, and I've ridden thousands of miles without one, but they're nice to have.
No one mentioned this, but I consider a cell phone a necessity. We had someone on a group ride have a (fortunately) minor heart attack and were glad we could call for help. Also very useful for less dramatic events (my wife loved when I called for a pickup when my chain failed). In the end, very few problems that can't be helped by a timely call.
And most phones come with an 'off' button if you don't want to receive calls.
One other item that is useful where I live is a bicycle train pass. If caught in a bad storm I can usually get to a train line that can get me close to home.
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