I have read that there are plenty of elementary particles. Can some of them form some species other than proton and neutron that are able to form a stable atom together with an electron or an electron analog (based on the definition that an atom is a relatively stable species constituted of a definite arrangement of some more fundamental species and has distinctive physical and chemical properties). Hence, can there be a whole set of periodic table of elements constituted of something other than electron, proton and neutron? You can assume I have no knowledge in particle physics (since I only studied up to high school physics). Google does not really yield any relevant result. Forgive me if it is a silly question because it is only out of pure curiosity.
However, it very much depends on what you mean by atoms. For example - as ACuriousMind above mentioned, one of the more common ones is muonium. This is a muon bound to a proton (instead of an electron bound to a proton, as in hydrogen). You can solve this particular model quite easily using the same methods used for solving the hydrogen atom (i.e. simply applying the Schrodinger equation for a Coulomb potential). Further, there is another rather common "exotic" atom called positronium, which is an electron bound to a positron, rather than a proton. You would think this would simply self-annihilate, but it does have stable energy levels! (Of course, annihilation does happen very quickly, but not as quickly as you might think!)
On the other hand, there is a vast array of exotic "atoms" that are more commonly found in high-energy particle experiments. In fact, because we cannot detect individual quarks (due to confinement), all of our results come from these exotic atoms. One such (well known) example is "charmonium." This is a bound state between the charmed and anti-charmed quark, "orbiting" their barycenter, with an energy spectrum remarkably similar to that of positronium. You may ask how these are like atoms - well, they are electrically (and strongly!) neutral, they have orbitals in much the same way (angular momentum, spin angular momentum, etc.), and, while they don't live long enough to "chemically" bond with other "atoms," there's no doubt that they are just a generalization of positronium, which is essentially a hydrogen analogue. The only difference between these exotic quark atoms (mesons and baryons, as they are called) is that they are not bound by the electromagnetic force, but by the strong nuclear force! It makes their dynamics significantly more interesting, but also significantly more difficult to understand.
I should specify - these are all "stable" in the sense that they form bound states. However, nothing besides "everyday" matter is stable in the sense that it lasts a significant period of time - if other substances did, they would be a lot more common.