It is surprisingly obvious yet one wonders what has contributed to its longevity and its countinued appearance in various arguments.
Why obvious? Well, let's consider the premises. Unless one is predisposed to believing that animals simply appear, then we must reject the premise that a chicken can exist as a fully formed adult without any previous existence. As a result, the only element left to examine is the egg. Yet, an egg doesn't spring into existence fully formed either.
However, the egg is a pre-requisite to any egg-laying animal, so we could conclude our answer to the question at this point already. Whatever else one might think, an animal must be born to become an adult, and an egg-laying animal is hatched from an egg to become an adult. Therefore an egg must precede the adult. What gives rise to the supposition that there's something more to it?
The problem is that the question operates under the flawed premise that species of any kind are absolute and that there are absolute cut-offs in how they are classified. It is important to remember that a species is an arbitrary designation based on how we elect to classify particular animals. If we were to examine the continuum of a chicken's evolutionary path, we would begin with some ancestor, and observe gradual variations occurring, until such time as we arrive at the animal we now call a chicken. However, what we couldn't do, is to draw a definitive line during this evolutionary path and declare everything on one side of that line as pre-chickens, and everything afterwards as actual chickens.
One possible way we could approach such a situation would be to use something like a genetic map to specifically define what we consider to be a chicken and then simply examine the genome of each creature until a match occurred. Obviously this match would've occurred in the egg [using our philosophical question], so we can definitely consider the egg to have occurred first, but it is equally important to note that our selection of a genetic map is also quite arbitrary. We, as observers, have set the definition, but it would be erroneous to conclude that some absolute dividing line had been crossed beyond our arbitrary definition.
An important point that is often overlooked, is that species designation often implies a directionality to evolution, as if there is some force that is compelling natural selection in the direction of producing a particular species. Yet, every creature alive is already a complete animal. Every one may well possess traits that are important, and given the right environmental pressures could prove themselves to be the ancestor of some future species. Of course, there is no way to know that presently, but it is important to recognize that there are no "transitory" species. There are no "missing links". Each animal is a continuation of its predecessors, and if sufficient changes accumulate, then we may consider them to be different species. However, that is an imposition of our rules and not nature's.
So we would find that these distinctions we use for classification aren't absolute and are only apparent when comparing these species over a sufficiently large number of generations where we can clearly see which is the ancestor and which is the modern day animal.
In the end, the point is that each animal lays an egg, and the animal that emerges is similar to the parent with the possibility of possessing a minor variation (1). So, strictly speaking it isn't necessarily an exact replica of the parent. As this process continues over many generations, we find that as each offspring varies slightly from its parents, the difference between it and its more distant ancestors becomes increasingly pronounced. Our designation of a species is little more than arbitrary divisions to account for varying degrees of evolutionary divergence. Consequently we can safely say that the egg always came first, albeit through a long ancestral line of eggs if we are to be precise (2).
(1) This is in keeping with the slow evolutionary path generally considered in natural selection. Clearly events and the environment may accelerate changes due to a variety of other causes, but the essence of the argument would remain the same whether the changes occurred over hundreds of generations or over one. The egg must always come first.
(2) Some arguments have tried to claim that this question is subject to the specific definition of an egg, in whether one means that it is intended to produce the chicken, or it is the first result of an actual chicken. This argument fails, because it presumes prior knowledge regarding the animal that laid the egg. Consider that if one were to simply receive an egg and analyze it genetically, if it contained a chicken, then it would be considered a chicken egg, regardless of the animal that laid it. Again, the egg is a pre-requisite to the existence of chickens regardless of how one wishes to define the terms.