Does infinity exist in the structure of physical systems? Sometimes people fail at asking a question by being too broad, unclear like here. So I'll take a stab at what I believe to be the same question, but more concise and clearly stated:

Does infinity exist in the structure of physical systems?

To be clear I'm referring to, systems in the real world, NOT models of systems.
Can the mathematical concept of infinity have any real connection with reality?
Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?
I have heard that if you model a physical system (recently Brian Greene posted a video on YouTube regarding infinity), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong.
 A: Infinity does not exist in physical systems in the same way the number 2 does not exist in physical systems.  Both are a mathematical abstraction, not a physical entity.  Both concepts appear when we start mapping physical systems into mathematics.  This mapping is often called a phase space.
Now that being said, we can ask ourselves questions about what mathematical spaces our phase space maps into.  By far the most common in physics is to map into the real numbers.  In such a case, we can now speak to a difference between infinity and 2.  2 is a real number, and thus something can map to it.  Infinity is not actually a real number.  It is a limit of the real numbers, but not a real number itself.  Mathematically, it acts just different enough to fall outside the definition of a real number.  So if one is mapping into real numbers, it simply doesn't exist, other than as a limit.
Now there's gotchas here.  There's calculus, in which we do operations on so-called infintessimals, which are smaller than any real number (the beauty of calculus is being able to do that).  There are also some cases where we might map to something other than reals.  The projective plane is one such mathematical construct.  However, they are certainly less popular than the real numbers.  Real numbers have a lot of really nice properties which get lost when we start using the projective plane, so we typically only use them when there's a strong advantage to doing so.
But if I may go back to calculus in summary, consider one of Zeno's paradoxes.  A man (Achilles, historically) wants to run to the end of a football field.  We are confident that he can do so, but can he really make it there?  For to reach the end of the football field first requires him to reach the half way point on the field.  So there's a step he must do before he can reach the end.  And to reach the halfway point, he must first reach the quarter-way point.  And the eighth way-point, and so on.  So Zeno would argue that Achilles must undertake an infinite number of tasks before he can reach the end of the football field.  Surely this is impossible.
Zeno would argue that this was evidence that motion itself was an illusion.  Nowdays, it can be thought of as the inspiration for calculus and handling such infinite series' with grace and physically intuitive results (yes, he can reach the end of the field... we know this at a very young age).  Sidestepping such infinities does require being very precise about what infinity is.  Most people define it in a way which is a purely abstract mathematical concept, a "limit," and as such it "does not exist" in physical systems.  But the devil is in the details, for we can say such things about a great many mathematical concepts.  This really is the beauty of math.  It can use entities that do not exist in the physical world to reliably make true statements about objects that do exist.
A: One example that might answer your question is in quantum mechanics. Some quantum systems require infinite dimensional quantum mechanics to be described. Even in the simple example of a wavefunction, we describe these in $L^2(\mathbb{R})$, which is an infinite dimensional Hilbert space. Although we cannot measure infinite energies, or any infinite value for any observables, there is a fundamental part of the theory which relies on the Hilbert space being infinite dimensional. Namely, the commutation relations
$$[\hat{x},\hat{p}] = i \hbar.$$
This fundamental property has deep implications, such as the uncertainty relations for $\hat{x}$ and $\hat{p}$, which have been tested to very good precision in experiments.
Most relevant to your question: no finite dimensional Hilbert space can have the commutator of two operators proportional to the identity operator (the trace of the commutator is always zero in finite dimensions). So, in a way, we have experimentally tested that even very simple systems require an infinite dimensional Hilbert space to be described.
A: 
Can infinity have any real connection with reality?

Yes.

Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?

Yes.
It may seem I've contradicted myself, but the resolution is to understand that this applies to every mathematical object we use in physics. Mathematical objects are not physical: they are products of human imagination. However, with care, they may be used to construct models of physical systems: that's their connection with reality.

Does infinity exist in either terms of structure, parameters, or measurement in any physical systems?

Here's an example of measurement, and old-fashioned multimeter with kΩ scale ending at $\infty$ on the left. 
Now, you may argue that's not really infinity: just a resistance higher than the meter can resolve. But the same is true at the other end of the scale, where 0 represents a resistance lower than the meter can resolve. Even other numbers like "2" don't represent mathematically perfect, ideal numbers here. Infinity is a perfectly practical, useful abstraction here, just like the numbers on the rest of the scale.

I have heard that if you model a physical system (recently Brian Greene posted a video on YouTube regarding infinity), and you run into infinity as a solution, then you have either made an error in your calculations or your model is wrong.

In that sense every model of a physical system is wrong, since no mathematical object exists in reality. But some physicists get so lost in the math that they forget this. The way we teach physics as an abstract, classroom subject doesn't help. Students come away unable to appreciate the physics that's around them at every moment, and they often can't even solve the simplest real-word physics problems.
You might see if you can find Jack Schwartz's essay "The Pernicious Influence of Mathematics on Science" (I don't know of a link).
A: I think the important clarification is between infinity, as a very rigorously defined concept in pure math, and infinity as it is often used in physics. In mathematics, infinity is very well defined. You can distinguish between countable and uncountable infinities, etc.
In physics however, infinity is generally used in a more loose sense. Physics is a domain of approximations and idealizations. Thus, if a value is  orders of magnitude larger than another, you could effectively approximate the larger term as infinite to clean up a messy equation. Infinity is therefore used most often as a limit in physics to assume a quantity is very large.
However, there are areas of more mathematical physics when you need to be extremely careful how you define and handle infinities. Perhaps the most well known example of this is quantum field theory, which is famously rife with infinities that need to be dealt with via renormalization procedures. In quantum field theories we deal with divergent integrals which cannot simply be cancelled by subtracting another large "infinite term" because mathematically you can subtract two infinities and still end up with another infinity. Thus, complicated renormalization schemes are necessary to correctly remove these true infinities.  Another example of infinities in quantum field theory is the difference between having an uncountably infinite number of un-normalizable free particle states vs. a countably infinite number of basis states for your Hilbert space once you impose normalization.
There are also unresolved questions in QFT related to infinity. For example, you run into an infinite "zero point energy" when constructing the Hamiltonian for a simple free scalar field theory. This implies an infinite energy density of free space. We don't observe this, and this problem is referred to as the cosmological constant problem.
Essentially, when thinking about infinity in physics, the context is very important: is it just being used to mean "much, much greater than" or is it being used as a rigorously defined mathematical concept? Both are true and neither are mystical.
A: Infinity and her twin sister singularity are ubiquitous in mathematical models, from elasticity theory to cosmology. Distributions, starting with Dirac's delta, were introduced and developed in order to tame singularities so as to make them mathematically tractable, preserving and enhancing the predictive power of the relevant mathematical models. I suggest not to confuse the phenomena with the math that we employ to track them.
A: 
Does infinity exist in either terms of structure, parameters, or measurement in any physical systems?

Infinity does definitely not exist in terms of measurement. (And thus also not in measured values of parameters.)
I had a look through some of the books on my shelf and found only two rather short remarks on this in Gerthsen: Physik and Martensen: Analysis 1, both saying in their first chapter that when measuring a continuous quantity there will always be an error/the measured value will not be exact. Both do not elaborate further, but it's obvious, really: Measurement will always be made with finite accuracy, limited, amongst other things, by storage capacity for numbers, finite measuring times etc. (If you want to contest that statement, try writing down a Googolplex or Graham's number, for a start.)
Infinity does, however, exist in terms of structure or parameters if what you are referring to with this is the mathematical model. I don't think I need to give a particular example, they are more or less everywhere, not least because of the success story of infinitesimal calculus.
Case closed, I would say.
Addendum: Here, of course, I mean infinity in a mathematical sense. Cf. e.g. https://philosophy.stackexchange.com/a/52589. Everything else becomes a question about the use of language, about the term "infinity".
(And with a question like

Can infinity have any real connection with reality?

you'd be more at home on philosophy.stackexchange.com. Although, in my personal opinion, you'd be wasting your time.)
For example, I could talk to you about "die unendlichen Weiten der Pampa" ("the infinite vastness of the (Argentinian) Pampas", in my mother tongue) while not meaning mathematical infinity. What I'd be referring to in this would be very close to concepts used in asymptotics: The extent of the Pampas is much larger than typical distances I am thinking in terms of, like how far my eyes can see or how far I can walk in a day, and thus in what I'm telling  you maybe it isn't useful to put an exact number on it and I'll just figuratively call it infinite. Feels similar to using that $b$ is infinite or $a$ is zero, depending on what I need currently, in a model where $a \ll b$.
A: 
Can infinity have any real connection with reality? Or is infinity purely a mathematical concept just used by physicists as a convenient way to describe the very large, an approximation?

Infinity is indeed a non-physical concept and purely a mathematical abstraction.
(You can argue for actual infinity at the extremes of physics, such as the extent of the universe. But outside of this extreme case there is no discussion: infinity is purely a mathematical, not a physical, idea. In fact, such extremety considerations within physics would strictly speaking rather belong to the science of philosophy than physics.)
As you mentioned yourself, physicists will use the abstract concept of infinity to describe very large quantities. "Very large" is relative, so it all depends on the extent of one quantity compared to another. If the details of a phenomenon have no significance or effectual influence on the outcome of a scenario, then we would idealize such details away.
For human-scale mechanics on Earth, we can to all practical purposes idealize, say, the Earth as infinite in size. When considering bacteria, you can idealize the human body as infinite in size. When considering a balloon hitting a wall, you may idealize the wall as having infinite material strength and rigidness/stiffness. Even though all these idealizations are not perfectly true, the exact details would not influence the results from our scenarios and are thus irrelevant - such idealizations provide simplification in exchange for exact precision, and when exact precision has no practical influence, then the idealization is warranted.
Many similar strictly-speaking-non-physical mathematical abstractions are used frequently in physics and engineering. We idealise the incoming Sun rays as parallel, we idealise the surface of the Earth as flat, we idealise the entire planet Earth as just a $0$-dimensional point when dealing with astrophysical scales etc.
A: This is a difficult question, but I'll take a stab at it.
There are things, which once thought infinitely divisible, are now quantised. For instance, color. Color is now realised to be finite in possibility, due to there being only so many atoms and therefore only so many combinations of atoms that give colors.
It is difficult, therefore, after new ideas of quantisation from Quantum Mechanics, to find continuous values in physical terms. Even Space, or Time, have possibility of quantisation, and therefore no longer being infinitely divisible.
However, there still are places for infinities in physics. For instance, probabilities, in Quantum Mechanics, are continuous. There can be probabilities of 0.1%, 0.0001%, etc. Any one of these can have infinite decimal places. Something could have a 3.141592...% chance of happening. This PhysicsStackExchange Question agrees as such.
In QM, there are an infinite paths the electron could have taken, and the sum, or average, of these, leads to the final answer. QFT has a similar process, and is called renormalisation.
The Stanford Encyclopedia of Philosophy says: In a rather informal sense QFT is the extension of quantum mechanics (QM), dealing with particles, over to fields, i.e. systems with an infinite number of degrees of freedom. Link Here
Furthermore, it isn't impossible that the universe is infinite in size, or in age.
There are places for infinities in physics, both the infinitely divisible, and infinitely large, for now. Although some infinities in modern physics are labelled as need for improvement, like black holes, quantum gravity (as Brian Greene labelled), or renormalisation, but there are others, such as those I mentioned above, which aren't necessarily a problem. Infinities can represent physical things in physics.
A: Infinity exists in physical theory. There are infinite-dimensional Hilbert spaces, there are cosmological solutions that are infinite in space or time, there is the infinite divisibility of space and time as modelled by the continuum of real numbers.
However, there is always the possibility that the end point of the quantum revolution, will be an outlook in which everything about the universe is finite. So I think the answer to the question is: maybe.
