This idea is based on the assumption that space is indeed this large, and that matter is distributed in a roughly even manner. Both of these assumptions can be tested by measuring the temperature distribution of the cosmic microwave background radiation and data from WMAP suggests that this is the case. This is perhaps the least controversial theory that predicts a multiverse.
Multiverse Level #2
The theory of eternal inflation predicts that Multiverse #1 is one of many 'bubble multiverses'.
All theories of eternal inflation predict that there are other 'bubble universes' and string theory predicts that some of these will have different physical constraints to our own. This is the multiverse that Fenney et al. are searching for.
Multiverse Level #3
String theory and black hole thermodynamics suggest that Multiverses #1 and 2 are holographic.
The holographic principle uses string theory and black hole thermodynamics to show that the three dimensions we experience are actually projected from a two dimensional surface. This means that there are identical copies of Multiverses #1 and 2. String theory was devised to reconcile quantum mechanics and general relativity. It is not yet proven but provides unique predictions that we will one day be able to test for.
Multiverse Levels #4
String theory predicts that Multiverses #1, 2 and 3 could all exist on one 'brane' amongst many.
String theory states that subatomic particles can be modelled as one dimensional objects known as strings. All subatomic particles are modelled by different strings oscillating in different ways. Strings of larger dimensions are known as branes. String theory predicts that universes take the form of D-branes which are formed when p-branes collide. p-branes may be colliding all of the time, creating multiple big bangs.
Multiverse Level #5
Quantum mechanics implies that there are an infinite amount of #1, 2, 3 and 4 Multiverses, many of which exist in the same spacetime as our own, we are simply unaware of them.
The Everett, or many worlds, approach to quantum mechanics predicts that everything, even Multiverses #1, 2, 3 and 4, obey the laws of quantum mechanics. This means everything that's physically possible actually happens. Although this multiverse theory is often considered the most controversial, some argue that we have more evidence for the Everett approach than any of the other multiverse theories. I have written extensively about the Everett approach here.
Multiverse Level #6
Physicists predict that some of the intelligent beings in Multiverses #1, 2, 3, 4 and 5 will be able to create artificial realities inside of computers that will be nearly identical to our own.
A number of physicists and mathematicians have argued that we will one day be able to use quantum computers in order to create virtue reality environments that are indistinguishable from real life. We may even be able to create virtual copies of people that think and feel like 'real' people. This makes it possible that we are currently living in one of these simulations. I have written more about this here.
Multiverse Level #7
Beyond Multiverses #1, 2, 3, 4, 5 and 6, the ultimate multiverse is composed of nothing but mathematics and so multiverses of every conceivable shape exist.
As our understanding of the world has developed, mathematical models have become increasingly accurate. This has led some to argue that a complete mathematical description of the universe would be a perfect description. The idea that the universe is really a mathematical object is attributed to Ancient Greek philosopher Plato and is echoed in the intuitive idea that mathematical laws are discovered rather than invented.
Tegmark argues that if the universe is mathematical then "complete mathematical symmetry" suggests that universes of every possible shape must exist. This multiverse theory suggests that there are other universes that obey different physical laws to our own. Some may not obey the laws of quantum mechanics, relativity or string theory. Some universes are simply empty two dimensional triangles and some are shaped like cups of tea.
What does all this mean?
The idea that there are other universes may seem far-fetched, but all of these theories are either based on widely accepted physical theories such as relativity and quantum mechanics, or on theories that make unique predictions which we will one day be able to test.
The fact that people find the concept of a multiverse so bizarre does not make it any less likely to be true. We only have an intuition for the physics we needed to survive during the period in which we first evolved. Every time that we have looked at things which are larger, smaller, faster or more massive than that we have had to confront long accepted assumptions about the world. The concept of the multiverse is perhaps no less strange to us than the concept of a moving Earth was to people in the 1500s. If we are going to use science to confront the true nature of reality, then history has taught us that we should not expect mundane answers.
References
Feeney, S., Johnson, M., Mortlock, D. and Peiris, H., 2011, 'First Observational Tests of Eternal Inflation: Analysis Methods and WMAP 7-Year Results', Physical Review D, Vol.84
Feeney, S., Johnson, M., Mortlock, D. and Peiris, H., 2011, 'First Observational Tests of Eternal Inflation', Physical Review Letters, Vol.107