For me, it would be Hardlight.
Hardlight was my very first exotic and it reminds me of a time where Destiny was a completely different game. I remember going back to orbit after a strike and seeing a yellow square pop up on the screen for the first time.
It's not the greatest weapon, but when I think of Destiny, I think of my Hardlight.
What about you?
Edit: I'm glad to so many different answers and not just Gally. Great answers guys. Keep'em coming!
Edit 2: I wish Hardlight was a little better, like if it had Arc damage. But thatstill doesn't make it exotic. It would be interesting to see ricocheted bullets doing more damage that direct hits. Bouncing rounds around corners and getting kills would be amazballs.
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I probably would have to go with icebreaker. I've solo'd so many nightfalls because of that gun.