Is there a theory where there are (recursively) infinitely smaller particles? So I read that electrons are just points, with no mass, and furthermore, protons look like they have some "size" but that's really 3 "point-like" quarks. We first thought atoms were the smallest possible particle. Then we thought the same about protons, electrons, and neutrons, I think, until we found quarks. 
Is there a popular theory that assumes this pattern continues infinitely, with quarks being made of of some smaller things, and so on? Or otherwise, is there a popular theory that specifically predicts that this isn't the case?
 A: There is no such popular theory now, but I understand a somewhat similar idea was kicked around in the 70's, before the Standard Model was worked out. This was associated with the "Bootstrap model" of Geoffrey Chew. I am both too young and in the wrong subfield to give you a very detailed account of how this works, but here's a quick Wikipedia quote:

Geoffrey Chew and others went so far as to question the distinction between composite and elementary particles, advocating a "nuclear democracy" in which the idea that some particles were more elementary than others was discarded. Instead, they sought to derive as much information as possible about the strong interaction from plausible assumptions about the S-matrix, which describes what happens when particles of any sort collide, an approach advocated by Werner Heisenberg two decades earlier.

(from Bootstrap model)
However, this idea fell out of favor after QCD was worked out. Aside from practical issues, I think the relatively simple symmetries and few ingredients seen in the Standard model seemed to be at odds with this kind of idea. Unless those many lower-level particles somehow only combine to create fewer composite ones, something that is not seen at any other level of reality, we're just running out of room for simplification.
A: It has been suggested that quarks and leptons may in turn be made up of other particles called preons. The idea was motivated by a desire to explain some of the apparently free parameters of the Standard Model in simpler terms, however it has gradually faded from popularity basically because it doesn't seem to work.
One of the problems is that when you bind particles into a region of space of size $\Delta x$ the Heisenberg uncertainty principle tells us that there is an uncertainty in the momentum of the particles given by:
$$ \Delta x \Delta p \ge \frac{\hbar}{2} $$
Given that we know from the LHC that the sizes of electrons and quarks has to be less than around $10^{-20}$m the uncertainty in the momentum is so large that it's hard to see how a bound state could be formed.
A: When physicists talk about "particles", they mean that the size of an (extended) object doesn't matter for the level of the description they have chosen. In that case we can simplify the full dynamics of the object to the movement of its center of mass. 
We can, for instance, treat planets as "particles", when we calculate their orbits around the sun, but we wouldn't treat the Earth and the Moon as particles when we are talking about the tides and the tidal lock of the Moon to the Earth. 
It is therefor important to keep in mind that calling and treating a physical object, no matter what its actual size is, as a particle is a choice at the level of an approximating description, it's not a fundamental property of the object itself. 
Quantum mechanics complicates things somewhat because "the center of mass" is not a well defined concept any longer, so the classical mechanics treatment that came with the "particle" approximation, is gone. One can still apply the concept that objects that behave like as if they are "small enough" can be treated without taking their size into account. 
For atomic physics this means that the size of the nucleus can be neglected (in most cases) and for high energy physics, so far, the electron appears as an internally structure-less object, that can also be treated without having to worry about another length (and with that another energy) scale. 
Having said this, "elementary particles" are actually quanta, i.e. they are measurements on a quantum field. So when we are saying that photons and electrons are "point particles", what we are really talking about is that the quantum fields of the standard model are being described as continuous fields that have well defined values (even if they are not scalars) over a continuous coordinate space. Whether this is ultimately "true" is questionable, but at this point no physical measurement has invalidated this hypothesis, so we continue working with it (out of convenience and because we don't know what to replace it with). 
A: 
Is there a popular theory that assumes this pattern continues infinitely, with quarks being made of of some smaller things, and so on?

Not popular, mainstream physics assumes that the elementary particles in the table are really point particles and there is no internal level of complexity. 

Or otherwise, is there a popular theory that specifically predicts that this isn't the case?

This is supported by the Bell inequalities which exclude a deterministic underlying level , for local theories. Lorenz invariance also is a criterion that discards a number of proposals.
Still there are good physicists working on deterministic models as Nobel laureate  G.'t Hooft , who has also contributed on this site. See his answers on the subject of deterministic theories here.
