Question

Which word correctly completes this sentence?

Answer 1

The answer is: My neighbor asked my brother and ME to buy bread for her.

You can't say: "My neighbor asked I and my brother to buy bread for her", but you can: "My neighbor asked ME and my brother to buy bread for her"!

What is more: if you remove "my brother" there is only one posible answer: 
"My neighbor asked ME to buy bread for her."
 
That is why the answer for this sentence is: ME.

Answer 2

My neighbor asked my brother and ME to buy bread for her.
Normally it would be I but for this sentence it is me because the English language says that this sentence is grammatically correct.