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.
