I have heard that me gusta usually has a romantic connotation when referring to people (as opposed to just saying that you get along well with someone). What about me encanta? Does it always have romantic undertones or can it be used non-romantically to refer to a close friend or someone you deeply respect?
Personally, I would not use me encanta to refer to somebody I just respect (however deeply). I would use it, however, to refer to somebody I admire very much or something that I like very much:
Me encanta el trabajo de Bertrand Russell, creo que sus ideas son geniales.
Me encanta Cervantes, pues El Quijote es una maravilla.
Me encanta el cine, veo muchas películas.
Generally speaking, if you are encantado by somebody, it means that you admire that person in the sense that you like them very much, not necessarily in a romantic fashion. You might have the deepest respect for a person like Ghandi, because you admire his passion, but if you don't agree with his ideas, you'd probably never say that you are encantado by him.
In other words, you may respect somebody very much without considering them role models, but if you are encantado by somebody, that person is likely to play some kind of model for you: it is somebody whose example you'd follow.
The romantic point of view is also correct, but mostly in a context of two people who are very close and probably sharing an intimate moment (not necessarily sexual):
Me encantan tus caricias
A girl might say to her boyfriend me encantas in public (or a guy to a girl, or other combinations), perhaps in front of a group of close friend or relatives. For example, after receiving a birthday present during a family reunion, something like "¡Gracias! Me encanta el regalo y me encantas tú" sounds natural and would not make other people feel uncomfortable. However, saying me encantan tus caricias in front your partner's parents would probably end up being an awkward moment.
I hope that answers your question!
Contrary to Janoma, I've used the expression me encanta many times to refer to some girl I liked a lot, usually in confidence to some friend, as in
Esa mina me encanta. Mina is slang for girl/woman in Argentina. Me gusta would also be ok, but with a lot less intensity, to the point that Esa mina me gusta would not make much sense.
Esa mina me encanta is very similar to saying
esa mina me mata, but me mata has a more physical connotation than me encanta. Alguien puede encantarte por su forma de ser, pero el físico de otra puede ser matador (de ahí que te mata).
Esa mina me vuelve loco is another way of saying the same thing, that can mean slightly different things depending on the situation. She can volverte loco for her personality, for her body, or some combination of factors.