Similar concepts can be found in other philosophers, going back as early as St. Thomas Aquinas, but our story starts with the Age of Enlightenment. Philosopher David Hume, writing in the empiricist tradition of philosophy, argued that there were two kinds of knowledge: trivial knowledge based on the definitions of words, or the "relations of ideas;" and profound knowledge based on the content of our experience. The former, we would now call "analytic a priori" knowledge; the latter, "synthetic a posteriori" knowledge (don't sweat if you don't understand the terms used; an explanation will follow shortly).
This dichotomy had been assumed in all mainstream epistemology up until this point, but is most often associated with Hume, and as such is now known as "Hume's fork." It would scarcely be a few decades, though, until a philosopher came to question this dichotomy.
That philosopher would turn out to be Immanuel Kant. Kant, in his attempt to reconcile the conflicting epistemologies of rationalism and empiricism, argued that there was not one dichotomy, but two: the analytic-synthetic, and the a priori-a posteriori. In the first dichotomy, analytic knowledge is knowledge that an object has a property that is contained in the definition of that object. For example, to say that an apple is a type of fruit is analytic knowledge, since an apple is a type of fruit by definition. To say, however, that a particular apple is red, is synthetic knowledge. Apples are not red by definition; indeed, an apple can very well be green, and still be an apple. Therefore, to say that an apple is red, if true, would constitute synthetic knowledge, not analytic.
The second dichotomy deals not with the scope of knowledge, but with how you come to know it. A priori knowledge is knowledge from first principles, independently of experience. A posteriori knowledge, then, is knowledge derived from experience. Putting these four terms together, we can view Hume's fork through the lens of Kant: the trivial type of knowledge is analytic a priori, since it deals with the definitions of words and is derived independently of experience, and the profound type is synthetic a posteriori, since it is derived from experience and deals with propositions that have content.
Kant's insight was that these are not the only types of knowledge. Indeed, there are two other ways to combine the terms: synthetic a priori, and analytic a posteriori. Of the two, only the former is dealt with in Kant's philosophy, and many concepts that Kant brings up are considered to be synthetic a priori. Mathematics, metaphysics, morality, aesthetics, and politics are all topics Kant discusses, and all of them are considered by Kant to be knowable synthetic a priori.
For those who already know about these dichotomies, the preceding overview is likely to have been boring, but it was necessary for those who don't. In any case, thanks for sticking with me this far, and I hope I've provided enough context to answer the question now. I promise that it's about to get interesting.
So, how might we conceptualize analytic a posteriori knowledge; the type of knowledge that Kant himself never used? On the surface, this seems like a silly question; if something is knowable merely through the definitions of words, then it should also be knowable independently of experience; how, then, can analytic knowledge ever be a posteriori? This is precisely the argument that Kant gave, and, to this day, the majority of philosophers accept the validity of the synthetic a priori, but not the analytic a posteriori.
However, we can see that this isn't necessarily the case. Notice that Kant specifically aimed to separate the two dichotomies; as a result, nowhere in the definition of analytic knowledge is it stated that it is a priori. Therefore, if all analytic knowledge is a priori knowledge, this fact would be known synthetically, not analytically. This means that it is not necessarily the case, even if it is the case. How, then, can we conceive of such knowledge, if it does happen to exist?
The interesting thing is that, once we delve into this question, we find a hint of irony: a way of conceptualizing the knowledge that Kant himself rejected might be found in the works of Kant himself! Let's take a look at one concept that Kant holds to be synthetic a priori: the content of morality. Roughly, Kant's aim is to develop a moral philosophy from the concept of freedom, and what he finds is that the concept of freedom contains that of morality; in other words, morality is part of the definition of freedom, if freedom is to be a logically consistent concept (an idea later echoed by Hegel). It seems rather interesting, then, that he calls this synthetic; it might be saying something about the world, but if it's derived from the concept of freedom, then it must be analytic by definition.
Likewise, it's interesting that Kant calls it a priori. In order for the concept of freedom to have any content whatsoever, an agent must be presumed, and this agent can be free or unfree. The existence of agents, however, can only be known a posteriori (if you can prove otherwise, you would be recognized as the greatest philosopher of the 21st century, for giving a sufficient response to solipsism). Since morality is both analytic and a posteriori knowledge, one way to think of this sort of knowledge is to remember how Kant derives his moral philosophy.
We don't even need to get into abstract philosophical concepts on which not everyone agrees, though, to find an example of how we might conceptualize this knowledge. Consider the proposition P: "Water is a molecule composed of hydrogen and oxygen atoms." Is it analytic or synthetic? A priori or a posteriori? Well, let us first examine what the proposition is actually stating. Most people would agree that water is, by definition, the molecule H2O (actually, it's a dynamic aggregate of many variations on this molecule, such as hydronium and hydroxide ions, but this is good enough for our purposes). If this is the case, then P follows trivially from the definition; the molecule H2O contains only hydrogen and oxygen atoms, water is defined as the molecule H2O, therefor water is composed only of hydrogen and oxygen. It follows that P is derived from analytic knowledge, not synthetic knowledge.
So, is it a priori or a posteriori? Well, if it's a priori, then it shouldn't require empirical investigation to figure it out (by the definition of a priori). Yet, as it turned out, it did. Water was originally thought to be an element in itself; it was a relevantly recent discovery that it is composite. It took empirical investigation by the early chemists to discover that water was, in fact, the molecule H2O. Therefore, P cannot be a priori knowledge; it must be a posteriori knowledge. So, I've established that P must be both analytic and a posteriori, contrary to our intuitions that such knowledge cannot exist.
Of course, I anticipate numerous criticisms of this argument, and I might pursue them in another video. However, my aim in this video is not to prove the existence of analytic a posteriori knowledge, but rather to show how we would conceptualize it if it existed, which is my interpretation of Jason's question. Also, showing how to conceptualize it is a major hurdle to showing its existence, since one of the major hang ups people have with this type of knowledge is their inability to conceive of it.
I shall now leave you with a little taste of something slightly off-topic, but relevant nonetheless. What, precisely, does analytic knowledge mean? Well, it has to do with the definitions of words, but how do we know what the definitions of words are? This is something we learn by interacting with other people; in other words, we learn it a posteriori. The reader is left, then, with a choice to make: is knowledge of meaning analytic, or synthetic? If it's analytic, then we have another example of analytic a posteriori knowledge. But what if it's synthetic? That would imply that analytic knowledge is derived from synthetic knowledge. Put another way, analytic knowledge is merely a type of synthetic knowledge, and there is thus no analytic-synthetic distinction. The reader, then, is asked to comment on which horn of the dilemma they chose, and why they chose it?