It is believed that forces are mediated by exchange particles. This means that two particles "know" what to do upon encountering one another by exchanging this packet of information. There is then the question of how the object particle "knows" the nature of the packet. We might suppose that there is yet another particle exchanged between the packet and object particle to identify the packet's nature. Now that particle begets yet another ... and so on ad infinitum.
The basic idea of an exchange particle then, leads directly to the hall of mirrors. When faced with this, we have but one option ... to look away and do other things which might bring us a return on our efforts presently.
In fact, though the universe can and indeed must "sum up" and infinite number of steps to arrive at a finite result, we cannot in principle follow the summation. The logical rule is ...
at the same time directly interact with an infinite.
To do so would create a logically impossible result such as observing something which is infinitely small, large or distant.
If it is true that logic is the author of all that we observe, we may inquire as to what solution to the above problem it has outfitted itself with. And this is ...
To make every particle or property ... fundamentally identical
Considering the distribution of particles in the initial setup of the universe, there is a problem of finding the probablility of any given number of particles being in a given unit cube of space. This problem is intractable at the outset since there is no "most probable number" given the set of all integers. Therefore, the only option which will render a sensible universe is to put the cubes in first ... then ... assert by logical fiat that into each cube there will be exactly one particle (with some modifications given).
Thus, the solution to the problem of finite existing relative to infinite has as its formal solution ... quantization. When confronted with this, make everything one unit in quantity then ... 1/infinity = 1/infinity = 1/infinity = ... problem resolved. And we are presented with angular momentum restrictions, charge restrictions and a host of others some of which are not well understood.
That charge should be quantized is easy to deal with. What strength of charge should be assigned to a given particle? Should it be the present value of charge C or Cpn ... or perhaps ... Cp-n ... ? Clearly "n" could be any number and to make a viable universe we must have some number ... so 1 it is.
Now, we might assemble any number of charges in one place and make a big charge again ... but ... the universe doesn't mind this since its concern is charge per particle and not the subsequent disposition of those quantized charges.
There is a final mystery
The quantization of mass is not observed. While it is true that particles have discrete masses, there does not appear to be a regularity to those masses which is the signature of quantization. We have a spread of mass values without apparent regularity but highly constricted to apparent as yet unspecified limits. i.e. there are no fundamental particles smaller than an electron and none bigger than a very large proton (unless one accepts the dubious proposal that a "black hole" is fundamental).
In fact, if the above puzzle were solved as neatly as ... hmmmm ... spin statistics ... the cat would already be in the bag ;o) and the remainder of physics would consist of filling in the details.
Note that not everything needs quantization ... just whatever leads to a problem with finite-infinite. If a quantity can be calculated so as to allow a normal finite result, there is no need for an elementary quantum and we may find multiple values confined within a finite range.