Is this even a question? I mean ok there would have been a time when we had more limited knowledge and this would have been a legitimate question. But today?
Let's not kid around. The answer I instinctively blurt out (and have been doing so since I was like 7-10 years old, having been fascinated by dinosaurs at that age) is simple: the egg came first.
Rewind 65 million years. There are no chickens. There were, however, many reptiles such as velociraptor that did lay eggs. So we have a very vivid factual example of a period in the past, with eggs and not chickens. So eggs predate chickens. Clear, obvious fact.
So why do people still ask the question today as if it's still the causality dilemma it was during Aristotle's era? Are people that ignorant? Or just too lazy to think about it a little? Certainly it doesn't help that a majority of people in the western world were brought up reading the Bible, which gives the answer as chicken in Genesis 1:21. However, creation has no factual basis and this is only going to serve to confuse things.
Though.. maybe there is a depth to the chicken and egg question. What if by egg, we specifically meant a chicken-egg? That returns to a much more abstract debate.
I think we first have to define what a chicken-egg is. I mean, is it an egg that hatches into a chicken, or is it an egg that has to be laid by a chicken? (We'll also assume that by chicken, we mean Gallus gallus domesticus. I don't think that's too much to assume.)
So if we say that a chicken-egg is one that hatches into a chicken? This allows that the chicken-egg could have been layed by a different species of animal, and a chicken emerged from it, meaning that the chicken came first.
The latter leads to another non-question. If a chicken-egg has to have been laid by a chicken then it follows that the chicken has to have come first. Since you can't make a chicken without first having a chicken-egg. So the first chicken was hatched from a non-chicken egg.
Let's see. The chicken is said to have evolved from a subspecies of Gallus found in Thailand. So at some point, there must have been an egg laid by what we'd consider to be Gallus that hatched into what we'd consider to be Gallus gallus domesticus. And the answer to the question lies in what you decide to call that intermediary egg. Is that a chicken-egg according to our first definition of the chicken-egg? Or is it not a chicken-egg because it wasn't laid by Gallus gallus domesticus?
So ultimately it all boils down to definitions. And actually, I don't find myself caring that much which definition you use..
Although according to Erikson, Larson, Gunnarsson, Bed'hom and Tixier-Boichard et al, it could well have been a chicken-egg that preceeded the chicken.