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"The Universe is commonly defined as the totality of existence" but that is more of a philosophical position. Empirically we know (or believe) that it is of finite size some believe it is expanding others that is (or will be) contracting.

As we gain knowledge about distant places, our scope of area has expanded from Planet, to Solar system, to Galaxy, to Universe. In the same way we continue to find smaller and smaller building blocks, from Elements, to Atoms, to Quarks.

It is reasonable to assume, that the same gains in knowledge the removed the burden from Atlas, will find the edge of the Universe, and something that is beyond that edge.

What is the name of that which exists beyond the Universe?

Naysayers, who may claim that there is "nothing" beyond universe, are reminded that their beliefs are no less strong than, those who claimed that "nothing" was supporting the Turtle or the Elephant who like Atlas also supported the world. Lack of knowledge does not mean lack of existence.

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There is nothing beyond the universe. –  gerrit Apr 9 at 22:41
    
    
If there would be something "beyond" the Universe, that would be (part of the) Universe as well. –  Py-ser Apr 11 at 3:34
    
@Py-ser from developing answers it would seem your commment should be "beyond" the universe, that would be (part of the) Universe, where the first universe is lower case, while the second is upper case Universe. It seems that our galixy is in a universe, which is in a Universe, which may or may not contain other universes. –  James Jenkins Apr 11 at 11:25
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This question appears to be off-topic because it is about philosophy. What's more, it does seem to cause discussion rather than seek for an answer. –  Walter Apr 12 at 20:21
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By definition, the capitalized word Universe denotes everything there is, so even if we one day discovered we're just a part of a Multiverse, all the parallel universes of it would still be parts of the Universe as a whole, where Multiverse would just describe its nature. Or, if some yet undiscovered regions of it would defy our current understanding of its physical laws and constants as we know them, all of it would still be a part of the whole Universe. So it really doesn't matter, if beyond the known universe, there are regions of honey, milk and chocolate biscuits and all of it is carried by a giant tortoise. All of it would be the Universe, the physical universe as we can observe, the honey, milk, chocolate biscuits and the tortoise. The lot. All of it. The whole shebang.

Notice that I use capitalisation here, i.e. you can have more than one universe, but they're all a part of the Universe. Without capitalisation, it's just any domain, a particular sphere in physical or metaphysical sense, and only a part of the whole Universe. Sadly, this capitalisation is often neglected or used inconsistently, as is often the case with earth vs the Earth (the top soil vs the planet), sun vs the Sun (any star with planets vs our Sol), moon vs the Moon (any natural satellite vs our Luna), even galaxy vs the Galaxy (any galaxy vs our Milky Way). For example, observable universe is a sphere, a region of space, within the Universe.

The beauty of this naming convention is, that we already know the name for (but not necessarily of) everything there is, even if we don't know or can agree on what all that encompasses, or what laws govern some regions of space, time, or some other, yet unknown existence of it. It is universally true regardless of anyone's beliefs, even if they choose to call all of it by other names or attribute this existence to a sentient being, super being, or God. We're all a part of everything there is - the Universe.

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"sun vs the Sun (any star with planets vs our Sol)" Are you saying that a star without planets is not a sun? –  Andrew Thompson Apr 10 at 8:09
    
@AndrewThompson That's correct. A sun is any parent star around which a planetary system revolves. A star without planets isn't a sun to anything, so it cannot be called a sun. It would be like calling all men parents, yet not all are. –  TildalWave Apr 10 at 11:55
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Thanks! My 'learn item' of the day. :) –  Andrew Thompson Apr 10 at 11:58
    
I think your first statement is wrong. As I recall from a talk with Stephen Hawking from the 80s, universe has been redefined to be anything that we were in causal connection with in the past, are in causal connection right now and ever can be in causal connection in the future, pretty sure it was in this talk: youtube.com/watch?v=HKQQAv5svkk not sure what time at, will post in more detail if I rewatch it ;) –  DrCopyPaste Apr 10 at 14:15
    
@DrCopyPaste Please read all of my answer. You'll see that I'm not even discussing much how anyone defines "a universe", I'm trying to answer the question up top and explain what's the difference with "the Universe". A universe can be anything you want it to be. The Universe is everything it is, where adding to it that it also "was" and "will be" is just semantics. I would also like to remind you that Stephen Hawking is not necessarily even trying to explain (and by no stretch of imagination trying to define) the Universe, but a universe that has an effect on us in one way or another. –  TildalWave Apr 10 at 14:33
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It is far from empirically known that the universe is finite in size. The observable universe is certainly of finite size, but that is just a product of the constraints of our observation. We don't know for sure whether the universe is finite or infinite, bounded or unbounded. [Source] What we do know is that right now the universe looks very flat as far as we can measure.

As mentioned in other answers, the Universe has traditionally been defined as "all that is", though in practice in astronomy it is this particular "field" of space-time that we exist in. New theories in physics have brought with them the possibility that there may be other universes in this sense. The word we use to encompass all possible universes is "the Multiverse". There are different types of universes proposed, and you can read a good summary of them in the Wikipedia Multiverse article. From there you might read of Brian Greene and others that you might want to pursue further. For example, Brian Greene has a book titled The Hidden Reality: Parallel Universes and the Deep Laws of the Cosmos which might be of interest to you.

Note: There is a sense in which we can say the universe is unbounded in 3D as mentioned in the linked source, but there are other senses in which the bounded/unbounded nature of the universe is questioned but which cannot be adequately discussed in this space.

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