|MadSci Network: Physics|
That's an interesting question, and nobody knows the answer.
From your question, it seems you are already somewhat familiar with the Standard Model(SM) of particle physics. The SM includes all known "matter" particles (quarks and leptons), as well as the three fundamental forces that we have been able (so far) to study at the quantum level: electromagnetism, the strong nuclear force and the weak nuclear force (also the newly-confirmed Higgs field, which is a little outside the scope of this answer).
Like the rest of physics, the Standard Model is descriptive, in that it is an organised and (so far) accurate model of the particles and forces as we understand them now. As you point out, the different forces can be thought of as acting on different kinds of "charges". But you could also turn this around by saying instead that our concept of "charge" is our way of categorising particles by their observable properties (like how they interact with different fields). Here is a nice illustration of how the known elementary particles can be arranged along different axes according to their different charges (see the description underneath for explanation).
If you have followed me so far, you have probably noticed that the SM is incomplete. Gravity, for example, is not included in the SM, since we have not yet found a quantum description that works with the other known forces. Dark matter and dark energy are still mysteries, and neutrino oscillation is not part of the SM's mathematical framework. So particle theory needs to be extended "Beyond the Standard Model (BSM)" before we have a complete theory.
So could there be more types of charges? Possibly. Some BSM models (such as Supersymmetry) propose the existence of new, yet-undiscovered particles. Organizing a larger group of particles would require new particle properties to distinguish them, and in many cases this corresponds to new kinds of "charges" and forces that affect them.
Of course, no evidence of supersymmetric particles (or other new forces) has been found yet, and it is possible that we could eventually find a unified particle theory that does not require new forces (or charges) beyond the four we know. An important aim of the Large Hadron Collider is to look for evidence of BSM physics, and many of us are eagerly waiting to see what might emerge.
Try the links in the MadSci Library for more information on Physics.