Looks kinda like a poorly disguised poll instead of a question.
There are many parts about as important as the bike.
Since I can't expect to call someone for a pick-up should I suffer a failure, things like tube and pump become quite important.
If you're talking about carry along with you as you ride you shouldn't be riding without some way of fixing a flat. A good mini pump that will reach 100 psi or so without breaking or without a Herculean effort is important. A good pump like the Lynskey Road Drive in the large size (they have 3 size offerings the smaller ones will take a lot more pumping and at least the smallest one won't reach 100 psi and I doubt the medium one will either), the large size makes it easier to pump requiring less effort and less strokes. Then you need a good set of tire levers like Soma Steel Core levers, a patch kit and a spare tube. I personally like Park glueless patches but that's a discussion for a different time. And you'll need a seat bag to carry your flat kit and tools in.
Next I think would come hydration, for without hydration you can't ride real far. Get a good bottle, the Polar bottles are really good and offer a degree of insulation that can keep your drink colder longer than just a plain bottle, and they're not all the expensive, or if you really want insulated quality for a drink to stay cold a very long time consider a stainless steel bottle but these of course cost more like the Klean Kanteen 20 oz insulated bottle, or for more fluid carrying capacity the Avex 24 ounce Brazos Autoseal stainless bottle.
Then, depends on who you talk to, some sort of lights front and rear used even in the daytime; but for use to be effective in the daytime they need to be bright. In the daytime the only thing I worry about is the rear, I can see them coming in front of me but not so much in the rear, but others will argue this, and with people today texting and yakking nonsense on the phone they could drive a long ways without ever seeing you so you need something bright that flashes to get their attention. I really like the new Cygolite Hotshot Pro 150 tail light, this thing is incredibly bright at 150 lumens max, there isn't anything else on the market that exceeds this brightness for the price. For the front if you need a headlight get one that has a strobe function, then it can do double duty strobing during the day and seeing the road at night like the Cygolite Expilion 720 or the Niterider Lumina 750.
Next in order would be a mini tool, lots to choose from here depending on how mechanically inclined you are will depend on how many tools you would want to have along.
Next would come some sort of ID and insurance information, money just in case; there's a few other small odds and ends you could carry but that covers the primary stuff...I think!