Someone who can blow up faces and towers with no support from allies whatsoever, and who does not rely on an organized team combo. DPS Garen or Tristana are great for this. Olaf and Xin Zhao are also solid picks (Olaf can jungle if you REALLY want to be antisocial). Master Yi and Ashe are good but tough to 1v3 or 1v4 without tank/cc help.
Alternately you can Heimer and push cheese. Heimer is pretty independent and can farm fast enough to get aura items, wards, 400 ap and still be quite tanky (think 2 rod of ages, this guy gets tons of gold)