G.K. Chesterton did not write a formal argument for the existence of God, but there are sketches of one in his book Orthodoxy. This isn’t surprising since his goal was to defend Christianity. This is attempt to take his argument from the chapter Ethics of Elfland. One argument is: 1. Life is a story 2. All stories have story-teller(s). Therefore, 3. Our life has a story teller(s)
I find his defense of one intriguing, “Now, the mere repetition made the things to me rather more weird than more rational. It was if, having seen a curiously shaped nose in the street and dismissed it as an accident, I had then seen six other noses of the same astonishing shape. I should have fancied for a moment that it must be some local secret society. So one elephant having a trunk was odd; but all elephants having trunks looked like a plot.” (Orthodoxy, p 57)
His argument summed up is this: unlikely similarities indicate a plot, and nature has unlikely similarities, thus there is an ultimate plot. There is clearly some intuitiveness to this, since it is a version of a design argument.