The chemical properties of and element classification of an atom, i.e. composite structures made out of protons and neutrons bound in a nucleus by the residual strong force, is determined by the number of protons it has, but its isotopes depend upon the total number of protons and neutrons in its nucleus.
There isn't an obvious maximum number of isotopes. But isotopes with more nucleons (i.e. protons plus neutrons) decay because the residual strong force has limited range and can only bind so many nucleons.
Theorists have now calculated, at least in the case of isotopes with an even number of protons (i.e. how many even-Z nuclei are possible), how many different possible bound isotopes there are and have found that there are 4829 possible bound even-Z nuclei with 8 to 120 protons, of which we have experimentally measured masses for about a quarter of them. The atomic element with the most protons ever observed as of 2018 was element 118, called Oganesson (which is in the noble gas column of the periodic table even though it is a solid at room temperature), with a mass in atomic mass units of about 300. It has a half-life of 0.7 ms. This study didn't examine the number of possible isotopes with more than 120 protons.
Assuming that there are an order of magnitude similar number of odd-Z nuclei that are possible, and adding in the isotopes with Z=1 to Z=7, there are about 10,000 possible atomic isotopes with bound nuclei and 120 or fewer protons.
The model used also predicts the mass of each isotope, and for the 1244 isotopes for even-Z atoms for which experimental data is available, the root mean square difference between the experimental data and the predicted values is 1.477 MeV which is very good considering that the mass of a proton is about 938.3 MeV, the mass of a neutron is 940.6 MeV, the mass of an electron is 0.511 MeV (atoms are a bit heavier than the sum of the masses of their protons, neutrons, and electrons due to their binding energy). An atomic mass unit has a mass of about 931.5 MeV.
For example, Gadolinium with Z=64 in the middle of the range studied has an average mass of 157.25 amu which is about 146,478.4 MeV. So the model is predicting the masses of nuclear isotopes to a precision of about ten parts per million.
In all, about 3300 atomic isotopes (a.k.a. nucleotides) have been observed, of which 251 are stable (i.e. having no observed decays). Attempts to synthesize atomic elements 119 to 127 have so far been unsuccessful.
The mass table in the deformed relativistic Hartree-Bogoliubov theory in continuum (DRHBc) with the PC-PK1 density functional has been established for even-Z nuclei with 8≤Z≤120, extended from the previous work for even-even nuclei [Zhang et al. (DRHBc Mass Table Collaboration), At. Data Nucl. Data Tables 144, 101488 (2022)]. The calculated binding energies, two-nucleon and one-neutron separation energies, root-mean-square (rms) radii of neutron, proton, matter, and charge distributions, quadrupole deformations, and neutron and proton Fermi surfaces are tabulated and compared with available experimental data. A total of 4829 even-Z nuclei are predicted to be bound, with an rms deviation of 1.477 MeV from the 1244 mass data. Good agreement with the available experimental odd-even mass differences, α decay energies, and charge radii is also achieved. The description accuracy for nuclear masses and nucleon separation energies as well as the prediction for drip lines is compared with the results obtained from other relativistic and nonrelativistic density functional. The comparison shows that the DRHBc theory with PC-PK1 provides an excellent microscopic description for the masses of even-Z nuclei. The systematics of the nucleon separation energies, odd-even mass differences, pairing energies, two-nucleon gaps, α decay energies, rms radii, quadrupole deformations, potential energy curves, neutron density distributions, and neutron mean-field potentials are discussed.
DRHBc Mass Table Collaboration, "Nuclear mass table in deformed relativistic Hartree-Bogoliubov theory in continuum, II: Even-Z nuclei" arXiv:2402.02935 (February 5, 2024) (392 pages).
Another study (admittedly not a terribly credible one given that it is a pre-print on the Social Science Research Network by authors not affiliated with universities or chemical/physics research institutions) argues from a different perspective, that there can be no more than 137 possible chemical elements:
Using the particle-wave dualism of microparticles and the Bohr model of the atom, it is strictly shown that the maximum number of chemical elements in the periodic table cannot be more than 137. Since, starting from element 138, the speed of a 1S-electron when moving around the nucleus of an atom must be higher than the speed light in a vacuum. Therefore, Feynmanium (Z=137) is the last chemical element. It was also shown that a decrease in the half-life of chemical elements correlates with an increase in the 1S-electron relativism.
Volodymyr Bezverkhniy, Vitaliy Bezverkhniy, "The Speed of Light and the Number of Chemical Elements" SSRN (December 4, 2020).
A 2022 paper has questioned the reasoning behind this calculation, however.
Another study has predicted properties of atomic nuclei with up to 174 nucleons plus one with 184 nucleons.