In certain boss battles, if you die there is a chance that rather than going to
the black Game Over screen that shows Sora floating on his back, you'll be
taken to a white screen with two options. These options are:
"I Won't Give Up!"
--Selecting this causes the battle to continue with you in control of Mickey.
"It's All Over..."
--Selecting this is the equivalent of getting a Game Over and takes you to the
regular Game Over screen (the one with Sora floating on his back.)
As for which battles actually have a chance of causing Mickey to appear, there
are 10 battles in which he comes to save you. These battles will be numbered
from 1-10 to make it easier in future sections when referencing the battles he
appears in. These 10 boss battles are:
01-Shan-Yu
02-Thresh Holder and Possessor
03-Shadow Stalker/Dark Thorn
04-Cerberus (First Battle with Auron)
05-The Hydra
06-Volcanic Lord and Blizzard Lord
07-Storm Rider
08-Xaldin
09-Grim Reaper (First Battle)
10-Xemnas (First Armored Battle)
Keep in mind that if Mickey's saved you before, he might not save you again if
you die on another of these battles. As for the exact chances that he'll
appear, well...