Certain skills and cards may give you bonus green dice, as noted on their description. Some checks may also give you bonus dice if you have a related ability. Finally, reaching certain Insanity Thresholds will add permanent bonus dice to all your rolls. There is no limit to the number of bonus dice that may be added to a roll (if you run out of dice, note the results and roll the same dice again).
NOTE: The bonus dice do not contain a tentacle icon.