Has there been any serious scientific inquiries into answering this age old question?
Most theories about origin of life say that unicellular organisms were first and multicellularity (like chicken) evolved much later, so in that meaning egg (single cell)* was first. Also, almost all chicken ancestors lay eggs (primitive birds, non-avian reptiles, amphibians, fishes and so on...) so eggs existed much earlier than chickens.
*in the meaning of zygote. The development of chicken embryo starts in hen's oviducts, so it's not a single cell at the moment when hen lays it (Gilbert, 2000).
This is a classic case of "it depends on the context."
- The Egg: As Marta Cz-C points out, the amniotic eggs that chickens lay were inherited virtually unmodified from their theropod dinosaur ancestors. So when considering the type of eggs, the egg came first.
- The Chicken: From a systematic perspective, you must first have an animal that you would call a chicken before it can lay a "chicken" egg. So considering the animal, the chicken.
The question makes the false assumption that chicken and eggs are different life forms.
A life form at stage x is not a different life form than that very life form at stage y.
You might as well ask of which came first, a 7 year old kid or an 8 year old kid.
But they are the same thing and therefore this question is invalid.
Dunno about any serious scientific inquiries into the answer but I always thought the answer is egg. At some point the modern chicken has come into being as the progeny of two pre-modern chickens, however it had to be an egg before it could be a chicken and its parents couldn't have been modern chickens.
However, all that presupposes that you can draw a line in the evolutionary history of the modern (extant) chicken and say this is modern and what goes before is not. I dunno if that really can be done.
Edit: came across this guardian article: http://www.guardian.co.uk/science/2006/may/26/uknews
"Whether chicken eggs preceded chickens hinges on the nature of chicken eggs," said panel member and philosopher of science David Papineau at King's College London.
"I would argue it's a chicken egg if it has a chicken in it. If a kangaroo laid an egg from which an ostrich hatched, that would surely be an ostrich egg, not a kangaroo egg. By this reasoning, the first chicken did indeed come from a chicken egg, even though that egg didn't come from chickens."
And from Prof. Brookfield of the University of Nottingham:
The first chicken must have differed from its parents by some genetic change, perhaps a very subtle one, but one which caused this bird to be the first ever to fulfil our criteria for truly being a chicken; Thus the living organism inside the eggshell would have had the same DNA as the chicken that it would develop into, and thus would itself be a member of the species of chicken