Our lives have already been greatly changed
Can we put already with greatly like this :
Our lives have already greatly been changed
Sara...the guides for placing adverbs are quite complex, and vary with the type of adverb - time, manner, frequency etc.
In your example, you need to consider what 'greatly' is modifying. It is modifying the change but nothing else, so it has to go and sit next to 'changed' to make that connection clear.
'Already' is modifying the whole sentence, so it can go in one of three places - beginning, middle, or end.
Already our live have been greatly changed.
Our lives have already been greatly changed.
Our lives have been greatly changed already.