Let us think what full happiness would be like. This isn't just partial happiness, but it is a happy state involving nothing unfortunate for one, nothing unhappy. Full happiness need not be maximal happiness. It is prima facie coherent that one might be fully happy at t1, but happier yet at t2, though the lesser amount of happiness at t1 would not have to involve any kind of unhappiness at the fact that it is not yet t2.
Full happiness has both mental and extra-mental components. To be fully happy requires a certain level of awareness of the events that make one be happy. No one in a coma is fully happy. But purely subjective states are insufficient. Being loved by others is surely a part of full happiness, but thinking and feeling that one is loved by others is not enough. The falsehood in thinking and feeling one is loved by others when in fact they despise one is clearly something unfortunate for one. Both the subjective component and the objective are essential to full happiness.
Next, for the sake of the argument, let me assume that there is a finite number N such that there are at most N subjectively different conscious states that are possible to one of us. At this point, I want this assumption to be ambiguous between different senses of "possible" (practical, nomic, physical, causal, metaphysical, logical, etc.) For instance, it seems plausible that there is such a number if mental states supervene on brain states and there is a limit on the possible size of the brain of one of us (maybe brains just couldn't function—or at least couldn't function as our brains—if they were more than a light year across), since although an analog system like the brain possibly is can have an infinite number of states, states that are too close together would not be subjectively distinguishable.
Now, it seems to me that a part of the concept of being fully happy is that the state of being fully happy forever is desirable. Let us take that assumption.
I will individuate mental state types in terms of subjective difference (feeling hot and smelling wintergreen are subjectively different, but smelling synthetic wintergreen and smelling natural wintergreen need not be subjectively different).[note 1]
The following seems plausible: Every qualitatively normal human state—i.e., every state of the same qualitative type as our normal, everyday human states—is such that to be in that state forever would be somewhat unfortunate. When we find ourselves feeling really happy, we wish that the moment could go on forever. But in fact, in the case of normal human states, this would be unfortunate. The wish of the lovers to sit on the bench watching the autumn foliage forever might be romantic, but if a fairy froze the lovers in that subjective state for eternity, we would see the spectacle as deeply sad. We might see it as preferable to many other states, but it would not be a fully happy state.
Neither would it be a fully happy state for a person to oscillate, with or without a repeating pattern, between a finite number of normal mental states. Granted, if the person in the state may be unaware that she has already experienced the blissful state 10100 times, she may not feel any ennui in having the state for the (10100+1)st time. But remember that happiness involves not just a subjective state, but an objective one. It may or may not be good to unaware of the infinite repetition of states, but such repetition is itself unfortunate.
But if there is only a finite number of normal mental states (distinguished subjectively) possible to us, then anybody who experiences only normal mental states will either cease having mental states (due to death or coma) or will eternally oscillate (with or without a repeating pattern) between a finite number of states. Since it is unfortunate if happiness is not to last forever, the person who would cease to have mental states was not fully happy (whether or not she was aware of the impending end of consciousness). And the person eternally oscillating between a finite number of states is also undergoing something unfortunate.
Consequently, assuming what has been assumed above, such as that there is a finite upper bound on the number of mental states possible to us, it follows that full happiness is impossible to us if we are limited to normal human states. The sense of "impossible" here matches the sense of "impossible" in the claim that it is impossible for us to have more than N subjectively different mental states.
From the above, an argument could be constructed that our full happiness would require either a supernatural mental state (such as the vision of God) or our going through an infinite number of different mental states (e.g., due to unbounded growth in knowledge).
In either case, the following seems interestingly true: Full happiness is impossible as long as naturalism is true. This might yield a desire-based argument against naturalism if we add the theses that any rational desire is possible to fulfill, that the desire for full happiness is rational, and that if naturalism is true, then it is impossible for naturalism to cease to be true. This requires some kind of a physical causal or nomic sense of "possible".
The above is just a sketch. Working it out would require carefully examining the different modalities and trying to find one in respect of which all of the premises of the argument are plausible. Something like nomic modality might do the trick. But this is all left as an exercise to the reader.