There are 5 ways in which simplicity can be ascribed to God.
First, on the 1st level of matter. Again, there simplicity means that God is not composed of (a) material parts that interact according to (b) some laws of nature that define and condition God. God is not a body, nor is composed of matter and form.
Second, the simplicity of the 2nd-level Father-Son-Holy Spirit is described in an earlier post, where I discuss both the unity of the persons and the unity of the divine intellect, will, and power.
In addition, God is not a union of past, present, and future life plus timeless abstracta but lives in eternity — a simultaneously-whole and perfect possession of interminable life.
Third, the simplicity of the 3rd level of goodness is due to our inability to discern intellectually any distinctions within it at all. Goodness cannot be known or understood, only judged (by a wise man) to be good.
Fourth, there is the union of the levels themselves. This is a highly mysterious issue, much more so than even the mind-body union in humans.
Fifth, the simplicity of our classification of God. Thus, St. Thomas argues in (ST, I, 3) that God is not composed of (1) essence and concrete object, (2) essence and existence, (3) genus and difference, or (4) subject and accident.
(1) The concrete Son and the Father’s mental grasp of Him through the Holy Spirit as the abstract form or essence of the Son contemplated in thought are self-same. God’s nature is not individualized in matter, such that the same nature, such as humanity, can be instantiated in many parcels of matter, such as individual men, but subsists in itself. Each angel, for example, is as if its own species, and so is God even more obviously.
(2) For St. Thomas, the main difference in kind between God and creatures is specifically the aspect of God’s simplicity of the identity of God’s essence and existence. I agree that God is His own existence, but I think that even this admittedly stupendous attribute pales in comparison with the distinction between 3rd-level goodness and lower-levels being.
(3) God cannot be described univocally by any property he shares with another thing + another property by which He is different from it.
For example, it is permissible to say that God lives. Does it mean that we can place Him in the genus “biological organisms,” such that He will constitute a species on par with bacteria, bees, or humans? A bee would differ as an insect that feeds on pollen and nectar and stores both and often also honey; a human would be a “rational” animal; and God would be a “divine” life-form. I don’t think so. God is alive insofar as He has or is a soul or spirit, unlike inanimate objects. We may even attribute to him intellect, will, and power. But the way He exercises the functions of life is so different from how creatures do so that no classification of God is possible. “God lives” is true merely analogically.
Thus, “God is not related to creatures as though belonging to a different ‘genus,’ but as transcending every ‘genus,’ and as the principle of all ‘genera’.” (ST, I, 4, 3, reply 2)
(4) Every divine property is essential to God. If God is essentially omniscient, then He would cease to exist entirely, if omniscience were somehow taken away from Him. Conversely, if God ceased to exist, then omniscience could not be predicated of anything else.