Not much. There are usually not many complications from hives outside of the psychological distress.
Infection. Hives from an allergic reaction or medication can itch a lot, and all that scratching can break down the skin and allow nasty bugs to cause an infection. Also, hives may be a precursor to a more severe allergic reaction, one that could interfere with breathing and be life threatening. If you think you might be having an allergic reaction and there's rapid swelling of throat/mouth - time for 911!