i don't agree that 'despite eating the apple' is wrong. yes, 'despite' must be used with some sort of noun or noun-like construction - but
gerunds, such as 'eating' here, certainly qualify as noun-like constructions.
the advantage of including the possessive is that it allows a change of subject ('despite jim's eating the apple, joe still had plenty of fruit left in his lunch box'); if the possessive is not there, then the gerund necessarily refers to the same subject featured in the main clause.
at least one venerable source agrees with me that 'despite eating the apple' is perfectly fine:
http://www.bbc.co.uk/worldservice/learn ... tv41.shtml (scroll down to the bottom of the page for the resolution of this particular issue)
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i'm assuming that your original source of information is
this; we have no idea whether the poster who provided that information is in any way credible.