depends on voltage and how long you run it (heck "they" say you can run an amp though 30 gauge) ... though none of the parts on the arduino can sustain an amp for long
The following chart is a guideline of ampacity or copper wire current carrying capacity following the Handbook of Electronic Tables and Formulas for American Wire Gauge. As you might guess, the rated ampacities are just a rule of thumb. In careful engineering the voltage drop, insulation temperature limit, thickness, thermal conductivity, and air convection and temperature should all be taken into account. The Maximum Amps for Power Transmission uses the 700 circular mils per amp rule, which is very very conservative. The Maximum Amps for Chassis Wiring is also a conservative rating, but is meant for wiring in air, and not in a bundle. For short lengths of wire, such as is used in battery packs you should trade off the resistance and load with size, weight, and flexibility. NOTE: For installations that need to conform to the National Electrical Code, you must use their guidelines. Contact your local electrician to find out what is legal!
What is thickest gauge wire that can be inserted into Arduino headers OR breadboard? The standard I see is 22 gauge, however if I'm going to buy more wire I want the most amp for my buck! Can 22 gauge wire support up to 1 amp?
18 gauge wire is definitely too large. I have a 500 ft roll of 18/2 though that I used for controlling the lawn sprinkler solenoid valves at my house and I still have a LOT of it left, so I try to use it whenever possible. It *can* work in the Arduino headers, but you really need to swage it a bit and reshape it into a square. Filing it down slightly also works. The 24 gauge wire that is in CAT-5 cable kind of works, but is thin enough that it will often come out when you are just moving the board around.