Among the variety of views on the nature of hell and damnation, annihilationism is certainly a minority. However, I have seen somewhat of a rise in its popularity, along with universalism, presumably due to moral dissatisfaction with the eternity of hell and the theodicies for that view. For those who are unfamiliar, annihilationism, broadly speaking, is the view that the damned will eventually be annihilated. This contrasts with universalism in which everyone is saved (or eventually saved if the view allows for temporary damnation) and with the view that the damned exist forever.
There are indeed simpler refutations of annihilationism than that which I will present here, specifically ones from the immortality of the intellectual soul and God's justice (see Grenier), but the argument here is one that came about through sudden epiphany and is, from what I can tell, a novel line of reasoning.
Before I begin, I have some preliminary remarks. Since not all forms of annihilationism are the same, I will be arguing against any form which proposes that the damned are either annihilated immediately upon damnation or that the damned experience a temporary punishment after which they are annihilated. Furthermore, to be annihilated is not equivalent to merely ceasing to exist in the common sense. I cease to exist in the latter sense when I die, since I as a human being no longer exist, yet my act of existence may still remain if my soul survives with it, so there is a distinction between existing in a complete and incomplete state, where I can truly say that I cease to exist when there is that transition from the complete to incomplete state. Annihilation does not consist in an incomplete state of existence, but is the complete eradication of the being, everything from its form to its act of existence. An annihilated being is completely vanquished from reality; it is nothing.
The first and least controversial point to be made is that a just reward or punishment is proportional to its merit or guilt, respectively. This is because the essence of justice is proper proportionality, that one may receive what is in accordance with what is due. It is obvious that something like the death penalty would be an abhorrent punishment for stealing a pack of gum precisely because the punishment is not properly proportional to the crime (abstaining from the notion that this pack of gum is so special that the fate of the world depends on its preservation, or something silly like that). This will be important later in the article.
The second point to touch on is the nature of salvation. The term "salvation" is the noun form of the term "save," but in order to be saved, there must be something to be saved from. All Christians would agree that what we need to be saved from is death, the cause of which is sin. So, in order remove the possibility of death, one must remove the possibility of the cause of death, viz., sin. This cannot be done by a mere return to our original uncorrupted natures, since Adam and Eve not only sinned with an uncorrupted nature, but also sinned while in possession of preternatural gifts which constituted an elevated state of nature. If we cannot be saved by a return to a natural or preternatural state, it must be the case that we can only be saved by an elevation to a supernatural state, a state beyond the limits of all created nature (as opposed to preternatural, which is beyond the limits of original nature but not beyond the limits of all created nature).
Now, the only thing beyond the limits of all created nature is God since He is the author of all created nature, so given that man must be elevated to a supernatural state in order to remove the possibility of sin and death, the question then becomes this: "what relationship between God and man would remove the possibility of man sinning?" The answer lies in the fact that sin, and any moral qualification, is only said of rational creatures, beings with intellect and will, so given that sin is essentially an intellectual defect, it requires an intellectual cure.
To summarize what has been said, there is an intellectual relationship between God and man such that this relationship elevates man above the limits of all created nature and renders it impossible for there to be any disorders of the intellect and thus renders sin impossible. Now, since this intellectual relationship is beyond the limits of all created nature and is what elevates man, it is clearly the case that this relationship consists in man being perfected by God in some way, viz. that man is a passive recipient of actuality given by God, rather than man actualizing some passive potency in God. This newfound actuality in man is something that will bring his intellect into act, as it has been established that this is an intellectual relationship. The proper operation of the intellect is understanding, and as Aquinas says, "the intellect in act is the intelligible in act," precisely because an act is specified by its object, and the object of the intellect in its act of understanding is the intelligible species by which it knows, so this new actuality will consist in new understanding.
Something to quickly point out here is that if the condition for the impossibility of sin in an individual consists in knowledge, then it simply follows by modus tollens that ignorance is the condition for the possibility of sin in an individual.
If the actuality in man, given by God, which is beyond all created nature and which renders it impossible for him to sin is an intelligible species by which he understands, then the only possible candidate for this intelligible species is the divine essence (the reception of which is called the beatific vision). Firstly, because the divine essence is beyond the limits of all created nature. However, the most important reason is this: given that knowledge is the proper operation of the intellect, the intellect is perfected if it is brought into act and thus understands, and the greater the intelligible the more the intellect is perfected, but the divine essence is the greatest intelligible, so knowledge of the divine essence is the greatest perfection of the intellect, and given that the perfection of a being's highest faculty is the greatest perfection of a being, man's highest faculty being the intellect, then the greatest perfection of the intellect is the greatest perfection of man, and since it is ignorance which is the condition for the possibility of sin, then if man is perfected in his knowledge such that he now properly knows his ultimate good and is satisfied in it, there is simply no room for the ignorance that would cause him to look for his good in something else. Thus, sin becomes impossible.
What does any of this have to do with annihilationism? I apologize for the breadth of information here, but it is essential to the argument. Since sin is impossible upon reception of the divine essence, then the only way one could cease to have such knowledge is if God rejects the man and withdraws that knowledge from him, but God rejects no man; it is only man that rejects God, and that is impossible for man if he cannot sin. In other words, there is no guilt that could be proportional to God's removal of the beatific vision as a punishment, so such removal would be unjust. Therefore, once the knowledge is had, it cannot be lost, and man with said knowledge will love God above all forever.
This is important because my main contention with the annihilationist position is that the only guilt proportional to a being's complete annihilation would be rejection of God while in possession of the beatific vision, but given that I have showed that this is impossible, then annihilationism is impossible. I will elaborate on this in the subsequent paragraphs.
From what has been said, it is precisely because of the nature of the beatific vision that it is not had by the damned, otherwise something akin to universalism would be true which is contrary to annihilationism. Therefore, the damned are in a state of ignorance, although a highly qualified state. This is not to say that all ignorance is inculpable, it is rather the contrary. According to the conception of justice above, damned persons will be punished in a manner proportional with their sins given their degree of culpable ignorance, for it would be entirely unjust for any inculpable ignorance to factor into one's punishment.
Now, what again is annihilation? It is the complete and utter eradication of the being into pure nothingness. It is precisely that degree of ignorance, however small, which makes it impermissible for God to annihilate a damned person, because no matter how low in the depths of Hell a person is, the fact that they have this degree of ignorance means that it is still the case that they seek God as their final end, and thus seek their own goodness and fulfilment of being, but that they are just severely mistaken. Sin is not only the rejection of God, but in a way, it is a rejection of one's own being, because to reject God is to reject one's own goodness and the end for which one exists, and so the lower one is in Hell, the more they reject their own being. The only reason one cannot reject their entire being wholesale is because of that degree of ignorance, and so the only way a person could, per impossible, reject their entire being is if they did so while in possession of the beatific vision, and then the only proportional punishment to that degree of guilt would have to be annihilation, but since the beatific vision entails the impossibility of sin, there simply could be no guilt to which a proportional punishment is annihilation of the person.