One can say something similar regarding the Super-ego. The Super-ego is the moral law implanted in us by society and thereby also represents something infinite. The law of our culture constantly demands we obey our society's norms. It never leaves us. A healthy person can negotiate between these two impulses- but ultimately the task is never finished.
The infinite nature of this task probably is one of the reasons why Freud began to describe a third category of desire after the First World War. He wanted to understand why irrational violence existed. He concluded that it was the result of something called "death impulses." The desires of the libido are of course life impulses, whereas the unconscious desire for the death is the opposite. Since law and desire are infinite tasks, the human subject acts in such a way to be rid of them by willing their own death. This partially is fulfilled in the routine fulfillment of desire. For example, the Freud noted that the French refer to orgasms as the "little death." Nevertheless, much like salmon swimming up river to die, human have an impulse to eventually end themselves and ride themselves of these infinite impulses.
Though of course Freud was atheistic in his outlook, its interesting implications for the study of religion and theology. My mentor in high school once commented that Freud is Augustine minus God. How many of the world's theologies deals with the problem of the infinity of desire is an important and should be explored. As a result, I think we can clarify some points in Luther's theology.
Two of the most honest non-Christian religions in dealing with the problem of the infinity of desire and law are Hinduism and Buddhism. Both share a doctrine of karma, making the task of the law continue through several lives and in fact forever. In the same manner, the Buddhist claim is that desire is the origin of all suffering. Since desire never has permanent fulfillment and therefore lead necessarily to attachment and privation.
In both cases the ultimate release is the disillusion of personal identity. In the case of mainstream Hinduism, the recognition that "atman (the self) is Brahman (God/ultimate reality)." Buddhism prefers the opposite. The self is in fact illusion or nothingness, and the sooner we realize it the better. In realizing this or becoming "enlightened," we thereby free ourselves both from karma and from desire. That is to say, we free ourselves from being subjected to these impulses by becoming non-subjects.
On the other hand, Catholicism is a bit more problematic in its outlook. The claim of Aquinas is that the infinity of desire will one day be fulfilled by the infinitely desirable thing, namely, the vision of God. What about the infinity of the moral task? No problem. It will continue on in purgatory, and somehow be finished at some point- somehow. The point is that the infinite human impulse for mastery and self-justification is something that can be realistically fulfilled- rather than ended snuffed out.
How does this relate to Luther? As I read Luther, I see the New Testament's solution to this problem. Luther's solution is not fulfillment of desire and self-justification, but rather death and resurrection- first Christ's and then ours.
For Luther our infinite desire is in fact expression of our desire to master God. We try to master God through our good works and to justify ourselves. Desire is itself the impulse to master the other and therefore a will to be God. In the same manner, so is our will to fulfill the infinite task of law and to justify ourselves. So, both ID and Super-ego are in fact simply two aspect of what Luther called the "divine ambition."
Conventional means of resolving this don't work. Our desire goes on forever. In the same manner, we will never be able to justify ourselves. We cannot erase the bad things we do, even if we go on doing good works forever. Time ensures that they are never erased. Even if we are good people (which is impossible!), we might not be at some point in the future. So we must be ever vigilant. There is never an end to it.
Jesus as the true God-man ends all this. He gives us himself and therefore ends our desire to master him. He offers up his person as an infinite payment for sins and thereby ends our infinite moral task. Having surrendered himself to us and extinguished our need to master God, he re-creates us.
His death is our death, just as his resurrection is our resurrection. He kills us in the waters of baptism and incorporates the corpse of our old being into the new creation. In our new creation, our old desires and moral tasks are shrunk down to their proper size. They are shrunk to the size of the dimensions of our earthly life, and thereby are freed to use these impulses for our vocation in the created realm. This is the end of our divine ambition. This because now, the infinite law of God no longer hangs over us. It has been paid for by his infinite righteousness.
In this, we become happy creatures and not unhappy gods.