Why doesn't magnetic attraction increase infinitely when distance reduces I am not a physics guy, but I am from science background. However, I do not remember the famous explanation of this old school puzzle which our physics teacher used to ask. I searched on the internet but didn't find a discussion/explanation on this. 
Suppose two opposite poles of two magnets are approaching each other. They are attracting each other and thus their distances are reducing from each other. Theoretically magnetic attraction's relationship with the distance between the magnets is
$$
F \propto \frac{1}{r^2},
$$
which means that, the lesser the distance, the higher the attraction force is. If this is true, just a moment later eventually the magnets would attract each other at an infinitely large force (compared to their current attraction force) as soon as the distance further reduced and reached at a nano meter scale. So why they just gets attached to each other, without virtually smashing on each other due to this very large (infinitely) attraction force. This should happen even for fairly small magnets as well as the formula suggests. But that doesn't happen. Why?
 A: You're making a category error - you're comparing behaviour from two incomparable situations.


*

*The electrostatic force between two point charges scales as $1/r^2$, where $r$ is the distance between the positions of the point charges, and it increases without bound as they approach.

*Likewise, the magnetic interaction between two point dipoles also increases without bound as they approach; it has a more complicated directional dependence depending on which way the magnetic dipole moments point, but it scales as $1/r^4$.
In other words, point sources exhibit unbounded growth in the force they experience for both magnetic and electric sources. Your original post, however, asks about a separate situation:


*

*As you've noticed, the force between two extended distributions with homogeneous magnetization does not increase without bound.

*The correct comparison to that is an extended distribution of electric charge, i.e. a slab with a homogeneous volumetric charge density; this will produce a constant electric field at its boundary, and it will exert a bounded force on similar extended distributions of charge.
A: Magnetism is only a portion of the more general force electromagnetism. In the macro scale, only magnetism is evident, but as distances get smaller we have to take into account electric forces.

A magnet is just a collection of atoms oriented so that they produce a magnetic field. This field has a wide range (longer than electric forces) which is why its effects are more visible in the macro scale.
Since your two magnets have opposite poles, they attract. But as they get closer, they start to enter the range of electric forces.

Due to atoms having their electrons on the borders, electrons are the first to interact when it comes to electric forces.
As the two magnets approach, their electrons get closer and begin to repel each other because of the same charges. This electric force is equal if not much more stronger than the magnetic force.

That is why the two magnets don't "virtually smash" into each other but rather "attach" due to the balance of attractive and repulsive forces at play.

If you want to learn more, there exists a plethora of online resources covering Quantum Electrodynamics (QED) such as at Brilliant.org or Fermilab's YouTube channel
A: In real magnets (eg. a bar magnet), the pole is not exactly at the end. It is a bit inside. So, you can never put the poles of two magnets indefinitely close to get the desired infinite force, as you cannot put a magnet inside another.
(Actually, magnetic monopoles do not exist. But you can arrive at the same answer of the force between two magnets if you assume they do exist. And to get the correct answer, you have to assume that these poles are inside the magnet. The field lines inside a magnet is similar to that inside a solenoid, and not like that inside an electric dipole)
This is how field lines inside a magnet are oriented. Note that there are no monopoles. The magnetic field inside is strong, but not infinitely.

This is how the field lines inside an electric dipole are oriented. The field becomes infinitely strong near the poles. However, even if you get a real bar electret, the field will not be infinitely strong when you go near the pole, as you can never really reduce the distance to zero. Also (I am not sure about this), the poles of a bar electret might be little inside it due to crystal structure.

Also, the force between two magnets roughly goes as$ 1/r^3$, when the distance between them is large compared to their size. Check force between dipoles for more details.
A: To bring two magnets ever closer requires ever smaller magnets and these will have ever smaller magnetic fields. 
A: I think the best way to answer this question is to use gravity as an analogy, because it's so much easier to relate to.
Gravitational force is also inversely proportional to the squared distance between the bodies. Or more strictly, the squared distance between the centres of gravity of the bodies. If the two centres of gravity were infinitely close, then both bodies have no size, thus contain no matter, thus have zero mass.
Similarly, if two bodies have no distance between them then neither can contain a charge. There's nothing to host it.
