Barron's Grammar Workbook (which claims to be written for those wishing to take "the SAT, ACT, and more") says to use a possessive pronoun before a gerund, and gives the following example:
Her asking the question shows that she is alert.
Then it goes on to give two more examples, the second of which is said to contain not a gerund but a participle:
I hope you don't mind my intruding on your conversation.
I hope you don't mind me intruding on your conversation.
The first version conforms to the rule, and I bet the 2nd is fine in speech. But from a pedantic perspective (which the SAT and the GMAT presumably adhere to), is the second version correct? The present participle following a noun (phrase) functions like a relative clause, so what is grammatically being minded in the second version is not the action of intrusion, but the individual referred to by the pronoun "me."