In an infinite Universe, would we necessarily see multiple "Earths"? With my memory issues I can't be certain it was Sean Carroll, though I believe it was a video of him that explained that in an infinite Universe, we would inevitably see large scale patterns repeated, such as a whole solar system or galaxy. So, in theory, we could expect that somewhere in this infinite Universe another identical Earth would exist. Also, I believe, he (again, I may have the guy wrong) even mentioned that we would see copies of ourselves. Of all of this made perfect sense to me considering the immensity of 'infinity', but then, yesterday, I saw this video, which really blew my mind. A pattern that will NEVER repeat, even in infinity? No way. This still kind of confuses me. For example, wouldn't we still see large scale pattern repetition in an INFINITE replication of this pattern? So, my question is is it possible that the Universe could be infinite, but could also not repeat?
 A: This is to flesh out a point that is not covered very well by existing answers. Existing answers are saying, roughly, "infinite number of examples, plus randomness, generates all possible outcomes infinitely many times". I just want to point out that there are some details in the nature of the randomness that may steer the answer one way or another.
It is clear that you can have infinite sets in which no examples of something occurs, or just one example. For example, there are infinitely many odd numbers, but none of them are even. There are infinitely many prime numbers, but only one of them is even. So the mere existence of an infinite extent or an infinite number of examples really does not guarantee that any particular item will get repeated. This is why people invoke randomness. The randomness can serve to blur away the special properties which prevent a repeat, such as the special properties of prime numbers. The word "ergodic" is used for a behaviour which, extended sufficiently in time or some other parameter, will eventually encounter all possible states in a state space. Certain kinds of randomness can result in such ergodic behaviour. But other kinds do not. A computer programmed to randomly find numbers not divisible by anything other than themselves or 1 will spit out random prime numbers ad infinitum, and at most one of them will be divisible by 2.
We don't know if the natural world has the kind of randomness that would result in the answer to your question being "yes". It could be "no".
One may also add that the physical universe may in fact be finite in spatial volume at any given time. That is also unknown.
A: In a truly infinite universe with some element of randomness (such as could arise from quantum fluctuations) then anything that can happen will happen - and will happen an infinite number of times. So, yes, there would be an infinite number of earths indistinguishable from our own planet, as well as an infinite number of solar systems and an infinite number of galaxies identical to our own.
However, even in an infinite universe, the part of the universe that we can ever see or interact with is limited by the distance that light can have travelled since the Big Bang - we call this the observable universe. Although the observable universe contains trillions of galaxies and probably trillions of trillions of planets, this is still a very very small number compared to the total number of galaxies and planets that could exist. So the probability of there being a duplicate of earth in the observable universe is very very close to zero.
The Penrose tiling that is described in your vide link is a fascinating object, and it is true that it does not have any exact translation symmetries (although it does have rotation and reflection symmetries). But note that if you take any finite region of the tiling, there are only finitely many ways that an area of that shape can be tiled by the kites and darts. So the pattern of kites and darts in that particular region will be repeated an infinite number of times in the whole tiling.
A: In a spatially infinite universe you will get infinite copies of any finite region if the initial conditions are random (or otherwise repeat in the right way).
One can certainly imagine a universe that has one spot of initial conditions leading to our observable universe but surrounded by repetitions of some other pattern, which would make us unique. Most people would intuitively assume initial conditions that ergodically cover all possibilities because they are randomly and locally generated. The imaginary pattern is not a very plausible set of initial conditions since it contains a high degree of long-range order.
Whether initial conditions like the non-repeating tiling example are plausible is another matter: here there is long-range order, yet it is caused by selecting one tiling out of an infinite possible ensemble. It is very unlike how people normally think of initial conditions since randomness (no particular structure) is the simplest assumption, yet one can certainly imagine physical theories that demand very particular initial conditions to be valid. It is just that they feel suspiciously artificial.
A: Not necessarily.
As pointed out in another answer, it depends on whether the constraints bounding a given region are necessarily repeatable. Given an infinite range of possible quantum states, there is no reason to require it to be so. Or at least, one must then set one infinity against another and ask, which is the (infinitely) larger of the two? A poser guaranteed to confound those lacking deep expertise in the mathematics of multiple infinities.
The main problem from a scientific perspective is that the infinite nature of the Universe is not provable. We might eventually be able to demonstrate that it is finite, but any theory predicting an infinite Universe is open to the long-established and highly respected criticism that a theory predicting an infinity has simply broken down at that point (the infinite field strength and density at a gravitational singularity being a case in point). Thus, an infinite universe is metaphysical speculation, it is not real physics. We can happily populate it with Boltzmann brains and other absurdities. Replicant Earths, even. But, scientifically, it is all nonsense.
A: There are different kinds of infinity. In particular, relevant to the question, there are two infinities. One is the infinity of integers, and the other the infinity of real numbers. The number of possible random configurations of stuff in an infinite universe (and even in a finite universe) is at least as big as the number of real numbers. (Possibly bigger, e.g., the number of possible mathematical functions.) If we are talking about repeating copies of objects, I think it is reasonable to assume that each object takes up a finite volume of space of a size bigger that a specific volume. The number of such possible objects that can fit in the universe is the infinity of integers. Therefore there are not enough spaces for all possible objects, and therefore it is not reasonable to to expect any particular object to be reproduced. (I am talking about large objects like the Earth which was mentioned in the original question. Small objects like protons, neutrons, and electrons don't count here. The QM small objects have only a small number of possible states independent of position, momentum, and velocity.)
A: Regarding: "The main problem from a scientific perspective is that the infinite nature of the Universe is not provable."
My understanding about the nature of science is that nothing in science (except mathematics) is provable, nor is it expected to be believed to be true with certainty. That is a distinctive difference between science (as well as philosophy - except logic) and (most) religions which have beliefs/tenets which are believed with complete certainty.
