I contend that when I was a child, had I rushed into the room and cried out to my mother "The house is on fire and the flames are taller than me," she would have responded gently but firmly "Than I." As far as she was concerned, there were no choices to be made there, and her children certainly weren't going to be allowed to think so.
To my mother (and her children), this use of "than" is a conjunction, which means it is followed by a sentence-like thing, even though most of the words in the sentence-like thing have been left out.
He likes her better than [he likes] me.
He likes her better than I [like her].
The house afire is a case of the second sentence-like thing:
The house is on fire and the flames are taller than I [am].
There's no way you can get a "me" into that sentence-like thing; it just won't go.
But other people (not related to me) say, no, it's not a conjunction there, it's just a little old preposition like "in" or "to," so the "me" is just fine. I suppose they might be right, but if you're going to allow yourself to do that, you have to very careful with sentences like
He likes her better than me.
There's only one possible way to read that, but it's up to you to make sure that's what you meant and if you're in the habit of considering "than" a preposition, you may not have said what you meant. As an editor, I've frequently had to ask the author what was really intended and very often I find that it is not what was actually said. Thinking of "than" as a conjunction and putting in the left-out words any time there might be an ambiguity is good insurance against this.
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