Does either Quantum Field Theory or the Standard Model of Particle Physics predict the maximum number of particles or fields that can exist?

There is a limit on the number of flavors in Quantum Chromodynamics, behind that limit Color Confinement can no longer exist. The Beta-function that describes the interaction strength at different scales (at one loop) is:

$$ \beta(g) = \frac{g^3}{16 \pi^2} \left( - \frac{11}{3} N_c + \frac{2}{3} N_f \right) $$

This is negative for $N_f = 6$ quark flavors and $N_c = 3$ colors which leads to the confinement phenomena. A positive value would mean that quarks must interact more at very small distances, which contradicts confinement.

Another interesting aspect of this is the unitarity of the CKM matrix. If there are more quark families than what is presently known, the measurements should eventually show violations of unitarity of this 3x3 matrix.

It depends on what kinds of "Quantum Field Theory or the Standard Model of Particle Physics" you are focusing on.

Other than the familiar quarks+leptons+ gauge bosons+ Higgs particle sectors in the Standard Model, there could be other sectors that are topological, that could have extended objects like strings (Cosmic strings) or different kinds of topological defects, or anyon particles, or even anyonic strings (that are neither bosonic nor fermionic), for examples see Ref. 1 and Ref. 2. You can describe some of these topological objects by fields of higher form gauge fields, etc.

In general, you can imagine there are other Topological sectors that somehow couple to the underlying standard model in some way. And there is no limit but many many number of anyon particles that you can construct, and you can see for example Ref. 3.

It just if the mother Nature uses these topological sectors as fundamental as the known Standard Model, then the Nature must weave her puzzle in a non-contrived but elegant way.