Why can infinite quantities not be shown in an experiment or observed in physics? To modern physicists knowledge, there are no truly infinite quantities that can be shown with an experiment or observation. Time is not infinite, it had a beginning. Matter and energy is finite (otherwise there would be a giant black hole instead of an Earth). Space could be infinite or finite depending on the geometry of spacetime, but there is no way to measure or travel an infinite distance. The observable universe has a defined limit, the cosmic light horizon.
Some classical mechanics equations give infinite, but with the discovery of quantum mechanics, the infinite are revealed to have been due to probabilistic effects, not a true infinite quantity.
Why can we never observe infinite quantities?
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Why can we never observe infinite quantities?

All physical measuring/observing devices (including our own senses) are constructed from a finite number of parts with a finite number of states and carry out a finite number of processes in a finite amount of time. Therefore it is impossible to design or construct a device that can measure or observe an infinite quantity.
We might use a mathematical model of reality that predicts an infinite quantity in some scenario. This is usually taken as an indication that the model "breaks down" or does not apply to that scenario, because we assume that physical quantities do not exist in reality. But you seem to be asking about observations and measurements rather than existence, which is a different question.
A: Infinities in the set theoretic sense do appear in physics, as every physics theory is based on an infinite set like a manifold or a Hilbert space.
About infinite quantities, remember that the symbol $\infty$ does not have a rigorous definition. To make the question precise, you'll have to specify the axioms that you're using to define the infinity. It's possible that some physics model could use, say, the surreal numbers. But I think they'd only be used as an intermediate step like we use imaginary numbers.
It is technically wrong to say that classical field theory gives infinities. Something like $\lim _{r\rightarrow 0} \frac{1}{r}$ is undefined within the axioms of classical field theory. It's not predicting an infinity. It's predicting nothing because it's meaningless within the axiomatic framework.
A: Adding to @gandalf61's answer, although it is not possible to observe a result of infinite magnitude using instruments that can only access a finite number of states, we can observe null results.
Examples include zero electrical resistivity, which is a signature of superconductivity. You can actually measure the magnitude of the voltage drop across the superconductor and the result will be zero, to as high a precision as possible. This means the conductivity will be infinite, but you will never find an instrument which can actually measure the conductivity of the sample to be infinite.
Similarly, the phase change observed in the Michelson-Morley interferometer experiment is also zero, the null result complying with the absence of the aether medium. There are other instances as well, which I cannot recall right now.
A: Infinities usually arise in physics because we can only see a part of the picture. We can't see any edges where things stop or change, just the same rules extending as far as we can see. What should we conclude happens beyond the range of our sight? We could invent some sort of edge or boundary, but we have no information on which to base our theories, no way to estimate how far it is, or how it works, or what happens beyond the end. So the simplest assumption we can make is to say that the rules and phenomena we see, and which we know exist and work consistently, simply keep on going, without end.
We don't have any justification for saying it keeps going without end, either, but this isn't really a theory about reality but a way of patching over the unknown, the least-misleading unjustified assumption we can make. And since infinity is only filler for regions beyond the range of what we can observe, by definition we do not observe infinity in the places it is normally used.
That said, there is a more modern perspective in which 'infinity' is not just a word for extending what we know beyond the horizon, without end, but a mathematical extension of the numbers, a type of 'quantity' that obeys particular well-defined rules and relationships. There is no reason we know of that physics should be so well-described by the mathematics of Real or Complex numbers - it's just an experimentally observed fact about nature that it does. And so there is likewise no reason we know of why there shouldn't be any phenomena described by 'numbers' that follow the rules for infinity. And indeed, there are some mathematical models for infinities that do indeed seem to have deep relationships with physics and geometry. These include the projective and conformal infinities.
However, even in these cases we can usually avoid explicit infinities, because the mathematics reveals relationships between the finite and infinite in which they turn out to be the same thing. For example, we can model the projective plane by resting a sphere on the plane, and mapping each point of the plane to a pair of antipodal points of the sphere by drawing a straight line through the points and the centre of the sphere. Every point of the plane maps to a pair of points on the sphere, but the sphere also has points along the 'equator' that seem to correspond to 'points at infinity' on the plane.
We can therefore avoid talking about the plane being infinite by instead talking about the sphere instead. We can describe a physical theory that talks about the 'geometry of straight lines passing through the origin', projects them onto a unit sphere, and everything in our theory is finite. Mathematically, it is exactly equivalent to a theory where we project lines through the origin onto a flat plane instead, that has explicit infinities in it (the lines parallel to the plane). From the perspective of the sphere, there's nothing different or special about them - they're just the points along a particular great circle, like any other. And there's no reason whatsoever why physics can't use them. But although they can be interpreted as actual infinities, it's a lot more likely that physicists are going to pick a finite interpretation. So if there are actual infinities in physics, it's quite likely that we would pass them by without noticing them or realising what they were.
Physicists who study projective and conformal geometry often consider the theory to be much more elegant and symmetric than Euclidean geometry. For example, rotations and translations are unified (a 'translation' is just a rotation about a point at infinity), and the corresponding physical quantities like linear and angular momentum, mass and moment of inertia, forces and torques, are discovered to each be components of a single quantity, and the linear and rotational aspects of rigid body motion can be unified into a single set of equations covering both. Another example is the AdS/CFT correspondence, in which quantum gravity is found to be equivalent to a conformal field theory on the boundary 'at infinity' - one of the very few insights into quantum gravity we have. There appear to be some very deep relationships involved. And given that projective/conformal geometries can be considered to be the result of extending Euclidean geometry to include points at infinity, there is a strong argument that they are indeed real and important in physics.
A: Here's an example of an instrument that claims, on its face, to be capable of measuring infinity.

Note the "$\infty$" symbol at the left limit of the resistance scale. Note also, the needle is beyond the $\infty$ tick: the meter needs a bit of adjustment.
But how trustworthy would it be even after adjustment? How can I be sure the adjustment is correct? The last tick before $\infty$ is only $100 k\Omega$. Resistors with a million times that resistance are commercially available, and other objects have even higher resistance. So, $\infty$ here means "indistinguishable from infinite using this instrument".
Every measurement is like that. Suppose this meter reads $2k\Omega$. Does that mean exactly 2? Of course not. Even the best resistance measuring devices can't measure reproducibly to better than $10^{-11}$. In math, 2.000000000000000000001 is different from 2, but in physics it isn't.
