Basically agree with Lorikeet here. From a strictly prescriptive standpoint, "taller than I (am)" is 'correct', but general usage has made the shift towards "me" as the 'correct' choice, especially in conversation.
just ask yourself if it is possible to say 'he is taller than she' NO, it's her which is the object pronoun.
He is taller than she
is indeed the grammatically correct way to state this thought. However common usage has made "He is taller than her" the default for many variants. She is not the object of anything, 'than' here acting as a subordinating conjunction.
NOW, let's have some fun.....
Rick likes his video games more than me. (Rick likes his video games more than he likes me)
Rick likes his video games more than I.
(Rick likes his video games more than I like his video games)
Steve hit Betty harder than she. (Steve hit Betty harder than she hit Betty)
Steve hit Betty harder than her. (Steve hit Betty harder than Steve hit her)
But like Lorikeet said, common usage is changing most people's default to the object pronoun. As such, I'd probably end up teaching that both are 'accepted' but that the actual 'incorrect' grammatically construction is preferred.