Is there a limit to the number of elements that can exist? Why/why not?
In fact, this is one of the open questions in theoretical physics.
Each atom has a definite number of protons, neutrons and electrons. The number of electrons is always equal to the number of protons, so that each atom is electrically neutral. However, the number of neutrons does not have to be the same. Two atoms having the same number of protons but different number of neutrons are called isotopes.
As the atoms get heavier (namely, contain more protons and neutrons) so they become unstable; the heaviest stable element that exists in nature is the Uranium 238, whose nucleus contains 92 protons and 146 neutrons). Heavier elements do exist, but their nucleus is unstable (there are currently 118 known species of atoms, the heaviest - number 118 is known as Ununoctium, or eka-radon). It is known that heavier elements can exist (but were never seen directly in lab). Basically, we can't just jam together an arbitrary number of protons and neutrons and expect a brand new element to fall out. There are just certain numbers of nucleons that work. If a nucleus has too many protons, then it emits a proton or otherwise decays to even itself out. Same thing with too many neutrons. Furthermore, when the number of total nucleons just gets too high, being fermions they must occupy higher and higher energy levels. Eventually, they reach a point where it's just more energetically favourable for extra nucleons to be removed. Excess protons or neutrons are dripped away from the nucleus, and this is referred to as nuclear drip lines.