Well, in this poll i totally disregarded any of the actual people you meet, i was merely looking at the concept of the species.
Asari hands down. From a human perspective, they're kinda immortal and can share thoughts/emotions on such a deep level it's mindboggling.
The depiction of an Asari matriarch is probably 0% accurate compared to what they'd probably actually be if they'd existed, since such an enormous amount of wisdom, knowledge and expirience in one person cannot be grasped by our current even horizon i guess.
Also their way to breed and their interaction with other species is just admiring.
On top of that they're friggin Biotic-Badasses and hawt chicks in tight suits.
Edit:
@GundamSentinel
Since they're sapient and can reproduce themselves, i guess they count as a species. At least in my world view. You don't have to be out of flesh, bones and blood to count as living for me.
Just to clarify: Yes, that would mean i also include EDI/TIM (Depending on the language you play) as a living being.