This is what I tell everyone to get first when they get a new bike: Seatbag, to hold the following: Spare tube (maybe two) Small multitool Mini-pump or CO2 inflator Tire patch kit 2x tire levers That assumes you have bidons and cages.
Those six things should get you by for many miles and should get you out of any trailside emergencies.
As with anything though, make sure you know how to use them when the need arises.
For your own personal bike shop, I'd recommend a good floor pump with a pressure gauge, some high quality chain lube, and a maintenance book for DIY repairs.
Also a helmet should be entirely recommended as a required purchase, and probably front and rear lights.
Nothing hurts worse then getting hit by a motorist that "didn't see you" at an intersection.
bicycles.stackexchange.com/users/20165/Chris Lee
Bicycle Helmet Suitable clothing and shoes Cellphone
Whatever else you "need" depends on your mechanical abilities and how independent you wish to be.
bicycles.stackexchange.com/users/1584/Daniel R Hicks
I would go for a good floor pump, as I find it can be extremely important in preventing flats.
One big problem I see is underinflated tires.
This can cause flats and other problems like rim damage.
A good floor pump will make it not so much of a chore to ensure your tires are always properly topped up.
If you plan on leaving your bike anywhere except your garage, then a good bike lock is also quite essential.
If you have those things, then I would probably opt for replacing the stock tires.
Often the stock tires on bikes aren't great, and you would be able to get a pretty good set of tires for that amount of money.
Could easily save you from lots of frustration down the road.
bicycles.stackexchange.com/users/163/Kibbee
(I don`t have a definitive answer) .
I would concentrate on what is most frustrating when it fails when riding a bike: Tire and wheels maintenance:
Tire levers.
Good air pump.
Good tube patch kit.
Replacement tubes and tires.
A set of hex wrenches (check Imperial vs. Metric) After that, you can add whatever you feel will help you maintain your bike (specific wrenches, lubrication and grease,...)
Insipired by: http://www.bikeradar.com/gear/article/home-wrench-building-a-cycling-tool-kit-from-scratch-43564/ http://www.bicycling.com/maintenance/bicycle-tools/16-essential-bike-tools
bicycles.stackexchange.com/users/13517/Max
If you already have the usual tools and clothing you can always need more of the typical wear parts: Tires, tubes, chains, brake pads, chain oil, cables, pants …
Otherwise I’d start with a proper stand pump, mini pump (for on the bike), tubes, chain tool, hex keys, lock, bottles, helmet … Clothing is of course essential but hard to guess the right size for ordering.
bicycles.stackexchange.com/users/19529/Michael
If it's a quality bike, it will attract professional bike (part) thieves – which are everywhere, like bacteria – and you should invest in a quality U-lock and a set of good anti-theft skewers for the wheels and seat post.
bicycles.stackexchange.com/users/19862/TehMacDawg
Depends on the type of tour you are doing.
My 20 mile work commute requires panniers, cold and wet weather gear for the winter, lights for early morning and late evenings along with the common small items: 1.
Two spare tubes2.
Patch kit3.
Bike-specific multi-tool4.
Frame-fit pump5.
Emergency identification6.
Water bottle with water Regular ID, insurance info, cash, phone, clothing, keys, safety equipment, bike lock, shoes to fit my clipless pedals, and other items taken for granted are assumed to be on the list.
Ziplock bags for anything that can't get wet may also be added to the list.
I choose to use a GPS enabled cycling computer for courses and routing.
On my tour from Maine to Florida, I carried significantly more.
For closer to home touring, one can go a long way with little more than a patch kit and a pump.
bicycles.stackexchange.com/users/20178/Brian Beaudoin
Context:My bike shop includes free servicing as a standard with a new bike. I don't need such thing and live too far away. In lieu of waiver I can get €100 of stuff from the shop.  Best bike problem ever: I need to decide what to get for free form the bike shop...  Can anyone advise on what they would consider to be absolute essentials as a road/touring cyclist?
What are the absolute essentials a cyclist should own?
bicycles.stackexchange.com
<road-bike><maintenance><touring><accessories><tools>
bicycles.stackexchange.com/questions/31514
bicycles.stackexchange.com/users/13892/larryone
The absolute essentials a cyclist should own: seatbag, spare tube, mini-pump, tire patch kit, good floor pump, tire levers, replacement tubes and tires, chain tool, hex keys, helmet, bottles, lock, front and rear lights, brake pads, chain oil, clothing, ziplock bag, GPS and U-lock. A good floor pump and a repair kit with tools, lubricant and spares is a basic requirement for maintaining your bike. A helmet and front and rear lights are important for safety, while theft protection is essential if you plan to leave your bike outside the house at all. Stock tyres can be replaced  if necessary. Appropriate clothing (including cycle shoes if applicable) is also important. Emergency identification and a cycle computer are also useful, but the list depends on your intended use of the bike.
A good floor pump and a repair kit with tools, lubricant and spares is a basic requirement for maintaining your bike. A helmet and front and rear lights are important for safety, while theft protection is essential if you plan to leave your bike outside the house at all. Stock tyres can be replaced  if necessary. Appropriate clothing (including cycle shoes if applicable) is also important. Emergency identification and a cycle computer are also useful, but the list depends on your intended use of the bike.