By saying God is “infinite,” we usually mean that He is unlimited and unmeasurable. Unmeasurable, strictly speaking, could describe properties of other entities besides God. But it’s not a meaningful comparison to God. What can or cannot be measured is subject to the limitations of the measurer. From the perspective of a ten-year-old, the number of protein links in his own DNA is unmeasurable. That doesn’t mean his DNA is infinite; nor does it mean some other person or thing couldn’t quantify it.
Taking unmeasurable to mean “logically impossible to measure,” then it is synonymous with infinite, but, even defined that way, it’s not terribly relevant when it comes to God. My intellect may be “unmeasurable” from the perspective of an amoeba, but that’s not exactly high praise among other people. The same applies, more or less, to the idea of something being “unlimited.” Logic places boundaries on all things, but that’s not really a “limitation,” so this really comes down to the same concept of being “infinite.”
According to logic, there has to be a single “First Cause.” Rather than making something like God impossible, logic makes Him necessary. The point being that God is not “part of reality”; He is reality. That has to be understood in a careful context, however. C. S. Lewis once joked that pantheism—the idea that “all is God”—is not really wrong, just outdated. When there was no creation, there was only God. Now that God has created, some things exist which are not God, but He is still the ultimate source and foundation of their existence.
Another helpful point is that infinity, as regarding God, is not a property a being can have in some attributes, but not others. One is either entirely infinite, or he is not infinite at all. Consider, for example, divine attributes like omnipotence, omniscience, and omnipresence. It’s not possible for a being to be “omnipotent” unless that being is also “omniscient.” How can one have the power to do all things if they don’t also know all things? How can a being be omniscient and not be omnipresent—to know all things happening in reality, but not know some things in that same reality?
Authors who create comic book superheroes run into the problem of “selective infinity” all the time. If you’re strong enough to lift a building, you have to be durable enough to support a building. If you’re fast enough to outrun a bullet, you have to be able to think fast enough to not smash into walls. Those attributes can’t exist independently of each other. On a larger scale, this is how “infinity” works with God. To have any one infinite attribute means, by necessity, all of your attributes must be infinite.
The point is that we break God’s attributes into chunks like omnipotence, omniscience, eternality, and so forth, only because that makes it easier for finite humans to talk about Him. In reality, all of God’s attributes come down to the same basic source: He is the one and only ultimate and necessary being. He is literally the source of everything else. God expresses this idea in His own words, calling Himself “I AM” (Exodus 3:14). He simply is.
From that perspective, it is impossible for there to be other infinite or unlimited beings. Everything that exists apart from the only necessary being is the result of God’s creative work. Logically, every created thing must be different from God. So, it is not possible for beings to be truly “infinite” in the same sense as God, since they cannot literally be God.
Speaking of what might exist “beyond” God is literally to speak of things we couldn’t understand even if they did exist. We really don’t know what it means to be God. So, in some obscure, academic, theoretical sense, there might be “other” things happening in God’s realm of experience. But those things would be as far beyond us as jet fighters are to bacteria. But it’s literally pointless (and even a little dangerous) to speculate too much on such things.
We also need to note that God is a “being,” and describes Himself that way (Genesis 1:26–27). He is immanent and transcendent, but He is also personal. This partly explains why our reality can operate as it does. You can’t get something in a cause that was not present in the effect. In God, we see communication, unity, and relationship in the Trinity. Without those properties in the First Cause, we’d never see them in the creation.