What size wire do I need for a 100amp underground 300 ft run?
So for anyone finding this page, none of the answers are incorrect or be qualified electrician. please for any electric services needed find and ask a qualified electrician. 

Basically the breaker size protects the wire, So the breaker should not be higher amperage than the wire rating, however can over size the wire to meet voltage drop requirements. load calculations are for what minimum size panel/service needed but not the actual service rating if the breaker from the service side is a 100amp breaker feeding this then wire rated for 100 amp, etc. Most ampacity charts are up to 100 feet and go up one size of additional 100 feet. So, if using a 100amp breaker at the service side, then need to minimally size the wire accordingly and there are different rating for non burial, burial and in conduit  Aluminum sizes will be larger than copper. For that service would use URD Aluminum 4 conductor cable (need 2 hot, neutral and should have separate ground to shop, grounds isolated from neutrals in shop panel. for 300 feet for 100 amp rated service I would use Aluminum direct burial 1/0-1/0-1/0-1/0, the forth can be as low as #4 for the ground (but also in conduit, even if in conduit must still be rated underground wire and required by code also) Also note the size wire the breaker can handle, cannot cut strands to make fit, no no no, so check the breaker specs first. So if not really needing full 100 amp load, can go down smaller wires sizes, but use smaller breaker to meet the maximum amperage rating for that wire. so for example could use a popular 2-2-2-4 URD Aluminum direct burial wire, but would only use a 60 amp breaker for that distance, 80 amp if needed but could experience some voltage drop at high load, bad for running motors any length of time. Copper would allow smaller wire size, bigger wire still fit into breaker but is way more expensive. best to google look at a few wire size charts better contact an actual qualified electrician.