CK has a Moon-Mercury-Saturn T-square (we don't know his birth time, but because the alignment is so precise at mid-day, we know it is an operative configuration the whole day that he was born), and this alignment comes through again and again in the themes and tenor of his comedy.
We see it in a generally existential tone to his work, with many jokes about death, aging, and a sense of life as being mostly filled with suffering and banality. This is particularly related to his Saturn-Moon square which, when problematic, can be the "Life Sucks and then you Die" archetype (Jean Paul Sartre is born with the conjunction). It has something to do with the Saturn-Moon's tendency to meet hardship (Saturn) when trying to satisfy one's emotional needs (Moon). Often people with Saturn-Moon will have faced significant deprivation (Saturn) in their childhood (Moon), and this can be such a powerful imprint that it forms a kind of poverty consciousness. Interestingly, this archetype can manifest in a totally opposite way (as is often the case with archetypes), and it is also the combination you see in very emotionally mature (Moon-Saturn) people who have worked very hard (Saturn) to learn how to get their emotional needs met, and are able to take responsibility for those needs (rather than blame the world for not meeting them). But I digress...back to CK's comedy.
In one very funny bit we see CK describing a conversation he has with his seven-year-old daughter. His daughter asks him "Daddy will the Earth always go around the Sun forever?" And before he can catch himself he responds "Well no, at some point the Sun is going to explode." This of course is very upsetting to her, and she starts crying. In an attempt to comfort her CK adds "Oh honey, this is not going to happen until you and everyone you know has been dead for a very long time." This is not helpful, and CK realizes too late that he has traumatized his daughter with the confrontation with her own death, the eventual death of everyone she knows, and the eventual destruction of the Sun (CK's Sun-Pluto conjunction).
Note the many Saturn-Moon themes: the child's innocence lost, the child confronting death, the sad little girl, the failure to protect the child from the existential dimensions of life, the parent making mistakes. It is also clear in this bit as well as other parts of the film that CK takes his responsibility as a parent very seriously, and that he feels the burden of that responsibility as a great weight. This is all classic Saturn-Moon material. When it is more positive, Saturn-Moon can be the responsible committed parent archetype: the hard-working parent willing to endure great hardships to protect and provide for their children.
It is also classic that CK's conversation with his daughter would happen when she is seven years old. Seven years is of course a Saturn cycle, with Saturn moving ninety degrees every seven years. And it is classic for confrontations with the more existential dimensions of life to happen in coincidence with the Saturn cycle. The first Saturn square natal Saturn transit occurs around age 6 or 7, and it is very typical for a child to have their first really serious confrontation with the existential dimensions of the adult world at that age. The Saturn cycle can bring the experience of "forced maturation," and here we have a story of a little girl forced to grow up too fast.
Here is a great clip from one of CK's previous comedy films where he again explores some of the challenges of being a parent. There are many examples of CK's Saturn-Moon here, but at the end of the clip we can really see the full T-square with Mercury coming through. CK is trapped in a conversation (Mercury-Saturn) with his child who won't stop asking him questions. Mercury-Saturn can be the negative thought loop, the being stuck in a constantly repeating phrase, the experience of being unable to solve an intellectual problem. In the course of answering her questions he makes reference to many Saturn-Moon themes: his being neglected by his own suffering parents, his parents lost youth, the fact that we are all alone in the universe, that no one cares about us, and the inability to solve the ultimate existential questions of life.
CK does a lot of comedy bits where he is making fun of his own stupidity or someone else's, and Mercury-Saturn is the combination of critical self judgement about one's intelligence. With the Moon, it really fits how he makes humor out of describing what he sees as the stupidity of young children (and his own stupidity as a parent). In his new film Hilarious he describes a fight he has with his daughter who insists that Fig Newtons are actually called "Pig Newtons," and again he is trapped in a meaningless conversation with a child having a stupid argument.
If we combine the Moon with each of the other planetary archetypes we get all of the various dimensions of what it can be to be a parent. CK is particularly possessed by and focused on the Saturnian dimensions of parenthood: the challenges, the responsibilities, the existential hard realities, but of course, there are many other experiences that being a parent can bring. Parents with the Jupiter-Moon combination might be more aware of the joy laughter and emotional richness that comes with parenting. Someone else who has Uranus-Moon might be particularly aware of the unpredictability of children, their disruptiveness as well as their precocious trickster playfulness. The Neptune-Moon parent might be more aware of the spiritual connection that can exist between parent and child, the sensitivities of children, the boundary issues in families. The Pluto-Moon parent might be particularly focused on the power dynamics that can exist between parent and child, the shadow dynamics within families, as well as the primal intensity of love that drives the parent-child bond.
The gift that astrology, and the archetypal perspective generally, can bring is the awareness of when we are trapped in a limited archetypal perspective. We can come to realize that we are possessed by an archetype, and then we can do work to move beyond the limits of that particular god's point of view. As we move towards wholeness we are able to appreciate the full range of the human experience, and not limit ourselves to the projections of one narrow perspective. We may be born with a particular alignment (like CK's Saturn-Moon) that has a very strong hold on us, but as we work with that energy and integrate the fuller range of its potential, we are able to move beyond the possession state and own the fuller range of archetypal potential that exists within each of us.
Louis CK's birth data found on Wikipedia (no time, noon chart).
I don't know how long it will be there, but the film "Hilarious" is currently posted on YouTube.